A Redundantly Actuated Chewing Robot Based on Human

Article A Redundantly Actuated Chewing Robot Based on Human Musculoskeletal Biomechanics: Differential Kinematics, Stiffness Analysis, Driving Force Optimization and Experiment

Haiying Wen 1 , Ming Cong 2,*, Zhisheng Zhang 1, Guifei Wang 3 and Yan Zhuang 4

1 School of Mechanical Engineering, Southeast University, Nanjing 211189, China; [email protected] (H.W.); [email protected] (Z.Z.) 2 School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, China 3 School of Science, Qingdao University of Technology, Qingdao 266033, China; [email protected] 4 School of Control Science and Engineering, Dalian University of Technology, Dalian 116024, China; [email protected] * Correspondence: [email protected]

Abstract: Human masticatory system exhibits optimal stiffness, energy efﬁciency and chewing forces needed for the food breakdown due to its unique musculoskeletal actuation redundancy. We have proposed a 6PUS-2HKP (6 prismatic-universal-spherical chains, 2 higher kinematic pairs) redundantly actuated parallel robot (RAPR) based on its musculoskeletal biomechanics. This paper

studies the stiffness and optimization of driving force of the bio-inspired redundantly actuated chewing robot. To understand the effect of the point-contact HKP acting on the RAPR performance,

Citation: Wen, H.; Cong, M.; Zhang, the stiffness of the RAPR is estimated based on the derived dimensionally homogeneous Jacobian Z.; Wang, G.; Zhuang, Y. A matrix. In analyzing the influence of the HKP on robot dynamics, the driving forces of six prismatic Redundantly Actuated Chewing joints are optimized by adopting the pseudo-inverse optimization method. Numerical results show Robot Based on Human that the 6PUS-2HKP RAPR has better stiffness performance and more homogenous driving power Musculoskeletal Biomechanics: than its non-redundant 6-PUS counterpart, verifying the benefits that the point-contact HKP brings Differential Kinematics, Stiffness to the RAPR. Experiments are carried out to measure the temporomandibular joint (TMJ) force and Analysis, Driving Force Optimization the occlusal force that the robot can generate. The relationship between these two forces in a typical and Experiment. Machines 2021, 9, chewing movement is studied. The simulation and experimental results reveal that the existence of 171. https://doi.org/10.3390/ TMJs in human masticatory system can provide more homogenous and more efficient chewing force machines9080171 transmission.

Academic Editor: Jan Awrejcewicz Keywords: chewing robot; masticatory biomechanics; redundant actuation; stiffness; driving

Received: 13 July 2021 force optimization Accepted: 15 August 2021 Published: 18 August 2021

Publisher’s Note: MDPI stays neutral 1. Introduction with regard to jurisdictional claims in A chewing robot in an oral context has been found in many applications regarding published maps and institutional afﬁl- the emotion and expression of social robotics, the analysis of the chewing process, dental iations. and joint prosthetic material tests and food science [1–5]. In order to design a high anthropomorphic chewing robot, the working principle of human masticatory system should be studied. Simultaneously, in turn, the chewing robot can provide a useful tool for studying biomechanical behavior of human masticatory system [6]. The masticatory system itself is Copyright: © 2021 by the authors. actuation redundant as the mandible is driven by greater groups of muscles than required Licensee MDPI, Basel, Switzerland. and is constrained by two temporomandibular joints (TMJs). The human mandible is This article is an open access article attached to the skull by muscles and guided by passive structures such as posterior teeth distributed under the terms and and TMJ at each side of the jaw [7]. There have been a variety of chewing machines and conditions of the Creative Commons devices available for replicating the human masticatory movements and occlusal forces. Attribution (CC BY) license (https:// However, the functions of the TMJ and redundant actuation features of the masticatory creativecommons.org/licenses/by/ system have not been given sufﬁcient consideration during the design of chewing robots. 4.0/).

Machines 2021, 9, 171. https://doi.org/10.3390/machines9080171 https://www.mdpi.com/journal/machines Machines 2021, 9, 171 2 of 22

However, the TMJ not only affects the mandibular movement, but also affects the force of the mandible. Ignoring the TMJ will result in poor bio-imitability of chewing robot. In the area of human-machine interaction, most of the researches [8–10] used one single servo motor to raise and lower the lower jaw. The problem with these robots is that they lack sufficient DOFs for reproducing complex human jaw gestures. Only a few chewing robots designed for robotic head have multiple DOFs, like Flores’s jaw [1]. However, the compound motions of this chewing robot were realized by simple combination of drive units. In the area of rehabilitation, Waseda-Yamanashi (WY) series of robots can imitate the movement of doctor’s hands to train the patients who suffer disorders in opening/closing the mouth [3]. The WY robot showed good performance in dental training, but the driving structure is not completely consistent with human masticatory muscles. The Waseda Jaw (WJ) robot was developed to cooperate with a WY robot for dental training and was also be used to explore the physical phenomena and resistance force during jaw training of patients [11]. However, the WJ robot used eleven artificial muscle actuators (AMA) which can only generate contraction forces. Aimed at quantifying food texture changes during chewing, a number of chewing robots have been developed [12–14]. However, either the TMJ was not considered or the relationship between occlusal force and TMJ force were ignored in the design of these robots. An in vitro wear simulator based on Stewart platform was developed to replicate dental wear formations on dental components, such as crowns, bridges or a full set of teeth [15]. Typical movements and occlusal forces can be realized on this simulator, but the influence of TMJ force to occlusal force was ignored. The wear behavior of condyle/fossa pairs in TMJ was tested in a mandibular movement simulator for an equivalent of two years of clinical use [16]. Nevertheless, the TMJ articulation force affected by occlusal force during human chewing movement cannot be replicated on this simulator. Moreover, as far as we know, there is no such mechanical device, which can reproduce the musculoskeletal actuation redundancy of human masticatory system and no literature has yet studied the relationship between occlusal force and TMJ force on a robotic simulator. Basically, actuation redundancy can be achieved in three ways: Actuating the passive joints, adding additional kinematic chains, or hybrid of the first two types [17,18]. That is, adding more actuators than necessary to control the robot for a given task [19]. Redun- dant actuation has been proved to be an effective method for eliminating singularities, improving manipulability, stiffness and torque distribution in parallel robots (PRs) [20–22]. Liu et al. [23] investigated a two degrees of freedom (DOFs) planar mechanism and analyzed the distribution of singularity of this mechanism. Zhu and Dou studied the kinematic model and Jacobian matrix of a 2-DOF 3-RRR planar PRs [24]. Marquet et al. [25] presented a 3-DOF and four actuators redundant parallel mechanism, ARCHI, which has three PUS chains and a PRR chain, improving the mechanical strength, stiffness and singularities. Wu et al. [26,27] compared the performance of a 3-DOF planar PR with its corresponding non-redundant one, showing the redundantly actuated one has better dexterity, littler sin- gular configurations and higher stiffness. It should be noted that most of studies are about planar type redundantly actuated parallel robots (RAPRs). So far, only a few studies are available about spatial ones. Kim et al. [28] added two redundant actuators to the Eclipse robot eliminating the singularities within the workspace. Zhao et al. [29] discussed the dynamic performance comparison between the 8-PSS redundant parallel manipulator and its counterpart 6-PSS parallel manipulator. It can be seen that these redundantly actuated robots are all achieved by the above three ways, but the redundant actuation of the robot studied in this paper is achieved by reducing the DOFs of the end-effector by inclusion of additional constraints. Bionic robot has been a hot research area for many years. To some extent, human body is a very successful evolved cyborg. Literatures about robots which are bio-inspired by human musculoskeletal structure have attracted much attention from the academic community, including mechanical design, stiffness analysis, dynamics modeling, and energy-efficiency analysis, etc. [30–32]. For human masticatory system, the maximum Machines 2021, 9, 171 3 of 22

bite force that human being is able to produce with one’s maximum strength can be up to 50–60 kgf [33], which is almost one’s own weight. While, the size of lower jaw and TMJ is very small to burden such a heavy weight compared with the leg and knee joint. Although the human masticatory system is relatively small, it has good performance in stiffness, force output and energy saving. It seems the musculoskeletal actuation redundancy characteristics and the existence of TMJs play significant roles in masticatory system. In the fields, such as man-machine interaction, implantology, pathology and masticatory biomechanics research [34,35], it is very necessary to understand how this nonlinear redundantly actuated system works and what role of TMJs play in human masticatory system. In addition, the relation between occlusal force and TMJ force during chewing is also a point of interest in academia. The hypothesis that the TMJ force increases with the rise of occlusal force has been widely accepted in recent years [36]. In terms of the principle of operation of the whole redundantly actuated masticatory system, there is still much work to be done. Computational and mathematical modeling, such as static model [37,38], dynamic model [39], and finite-element model [40,41], were utilized in studying the masticatory system these days. Various objective functions for solving this actuation redundant system were evaluated: Minimization of joint loads, minimization of muscle effort [42] and minimization the sum of squared muscle activations [43]. These computational models are beneficial to the design of chewing robot. Meanwhile, the chewing robot can trace their foundation back to the physiological structure and computational model of the masticatory system and in turn, modeling, analysis and experiments of chewing robot may also verify the hypothesis of computational models. In order to study the good performance of optimal stiffness and energy efficiency in redundantly actuated human masticatory system from a robotics perspective. Section2 introduces the bio-inspired 6PUS-2HKP (6 prismatic- universal-spherical chains, 2 higher kinematic pairs) redundantly actuated chewing robot which was proposed based on masticatory biomechanics. This robot can not only mimic the musculoskeletal actuation redundancy of human masticatory system, but also reproduce occlusal force and TMJ force more realistically by introducing two HKPs. In the modeling and analysis of this nonlinear system, the differential kinematics model of this RAPR is derived in Section3 . For this RAPR with mixed DOFs, dimensionally homogeneous non-square Jacobian matrix formulation method based on three end-effector points are proposed. In order to study the significance of TMJs in redundantly actuated human masticatory system, the stiffness characteristic based on Jacobian matrix is analyzed and the optimization of driving forces is investigated in Section4. The simulation results of the 6PUS-2HKP RAPR and its counterpart 6-PUS PR are compared to prove the RAPR with point contact constraints has better performance in stiffness and energy saving. Section5 presents the chewing experiment of the RAPR. The chewing forces of the robot are verified and the relation between occlusal force and the corresponding TMJ force are studied via chewing simulated foods.

2. The Masticatory System and Redundantly Actuated Chewing Robot 2.1. The Masticatory System The masticatory system consists of an upper and a lower jaw. The lower jaw is attached to the skull pivoted at the condyle via two TMJs. There are a large number of muscles of various shapes and sizes driving the lower jaw to perform chewing movement, among which the main muscles include masseter, temporalis and lateral pterygoid muscles [7]. Each of them is made up of multiple groups of muscle ﬁbers, i.e., the masseter muscle is composed of MAS-S and MAS-P, as shown in Figure1a. Different muscle with its line-of- action contributes to masticatory movement in a unique manner. The muscle insertion coordinates and angle of line-of-action of each muscle group are estimated by referring to [14,44]. The resultant force line-of-action of each muscle group (MAS, TEM and LPT) is indicated by the purple arrow in Figure1a. The TMJ is a joint between the temporal bone and the condyle of the lower jaw. As the joint is treated as a rigid body, the disk and ligament are ignored. The condyle moves anterior and inferior along the mandibular fossa, Machines 2021, 9, x FOR PEER REVIEW 4 of 21

referring to [14,44]. The resultant force line-of-action of each muscle group (MAS, TEM and LPT) is indicated by the purple arrow in Figure 1a. The TMJ is a joint between the temporal bone and the condyle of the lower jaw. As the joint is treated as a rigid body, the Machines 2021, 9, 171 4 of 22 disk and ligament are ignored. The condyle moves anterior and inferior along the mandibular fossa, enabling the mandible to accomplish the translation movement, as shown in Figure 1b.enabling The condyle the mandible also revolve to accomplishs on the the fossa translation surface movement, to complete as shown the inrotationFigure1b . movement. In general,The condyle mastication also revolves movement on the fossa involves surface to both complete translation the rotation and movement. rotation In movements at the samegeneral, time. mastication Different movement with other involves joints both translationof human and body, rotation like movements hip or knee at the joint, these two TMJssame are time. correlative Different, which with other are joints guided of human by two body, articular like hip surfaces or knee joint, linked these by two TMJs are correlative, which are guided by two articular surfaces linked by the rigid lower the rigid lower jaw jaw(see (see Fig Figureure 1).1).

TEM TEM-A TEM-P TEM-A Left TMJ TEM-P LPT-S Mandibular LPT LPT-S MAS-P fossa Right TMJ MAS-P LPT-S LPT-I Resultant Condyle center Fossa MAS force MAS-S point surface MAS-S Lower Condyle Z incisor point Trajectory of condyle

Y Movement X (a) Ventro-lateral view of the masticatory system. (b) The motion of TMJ. FigureFigure 1. The1. The composition composition of the masticatory of the masticatory system and TMJ system (MAS-S: and superﬁcial TMJ. (MAS masseter,-S: MAS-P:superficial deep masseter,masseter, TEM-A: MAS- AnteriorP: deep temporalis, masseter, TEM-P: TEM Posterior-A: Anterior temporalis, temporalis, LPT-S: Superior TEM- lateralP: Post pterygoid,erior temporalis, LPT-I: Inferior LPT lateral-S: pterygoid;Superior MAS, lateral TEMpterygoid, and LPT representLPT-I: Inferior the resultant lateral force line-of-actionpterygoid; ofMAS, masseter, TEM temporalis and LPT and represent lateral pterygoid the resultant muscles, respectively. force line- Only the main muscles are shown). of-action of masseter, temporalis and lateral pterygoid muscles, respectively. Only the main muscles are shown.) As there are more muscles driving the lower jaw than required, and two TMJs constraining the motion of the condyle, the human masticatory system itself is mechanically As there are moreactuation muscles redundant driving [45]. There the is a lower very short jaw distance than in required the direction, and which two the TMJs condyle moves perpendicular to the articular surface. The articular surface of the TMJ is assumed constraining the motionto be rigid of and the the condyle condyle, keeps the contact human with the masticatory surface all the time. system The DOFs itself of is the mechanically actuationlower jawredundant is four. [45]. There is a very short distance in the direction which the condyle moves perpendicular to the articular surface. The articular surface of 2.2. The Redundantly Actuated Chewing Robot the TMJ is assumed to be rigid and the condyle keeps contact with the surface all the time. Inspired by human masticatory system, a 6PUS-2HKP RAPR has been proposed [46]. The DOFs of the lowerThe major jaw challengeis four. in designing the chewing robot deals with the TMJs and the masticatory muscles. In human TMJ, the two condyles are constrained in moving along the fossa 2.2. The Redundantlysurfaces, Actuated as shownChewing in Figure Robot2a. Although a ball-socket joint has three rotational DOFs, it does not allow for translations. While, the condyle actually moves almost freely along the Inspired by humanarticular masticatory surface of the system, temporal a bone 6PUS in its-2HKP articular RAPR capsule. has Therefore, been proposed two point-contact [46]. The major challengeHKPs in are designing designed to themodel chewing the articular robot contact deals of human with TMJs. the The TMJs HKP consists and the of a masticatory muscles.condylar In human rod and TMJ, a fossa the structure, two condyles as shown in are Figure constrained2b,c. The condylar in moving rods keep along contact with the fossa articular surfaces. the fossa surfaces, as shownSix PUS in chains Figure are 2a arranged. Although in parallel a ball corresponding-socket joint to thehas resultant three rotational line-of-action DOFs, it does not allowof six musclesfor translations. (MAS, TEM While, and LPT the in Figure condyle1a) to actually mimic the mainmove masticatorys almost muscles.freely along the articular Thesurface determined of the angles temporal between bone each linkage in its and articular the vertical capsule. plane are Therefore, shown in Figure two1d. point-contact HKPsTaking are designed advantage to of themodel bi-directional the articular driving contact of the engineering of human solution, TMJs six. The PUS HKP linkage realizes both mouth-closing and mouth-opening functions. To avoid singularity (dotted consists of a condylarlines rod in Figure and2 d)a whenfossa the structure, lateral pterygoid as shown PUS chain in isFigure near perpendicular 2b,c. The tocondylar the sliding rods keep contact withrail of the the fossa prismatic articular joint, the surfaces. tilt angle of lateral pterygoid chain is adjusted to slightly Six PUS chainslarger are arrange than its angled in ofparallel line-of-action. corresponding Figure2e illustrates to the the resultant kinematical line diagram-of-action of the of six muscles (MAS, TEM and LPT in Figure 1a) to mimic the main masticatory muscles. The determined angles between each linkage and the vertical plane are shown in Figure 1d. Taking advantage of the bi-directional driving of the engineering solution, six PUS linkage realizes both mouth-closing and mouth-opening functions. To avoid singularity (dotted lines in Figure 2d) when the lateral pterygoid PUS chain is near perpendicular to the sliding rail of the prismatic joint, the tilt angle of lateral pterygoid chain is adjusted to slightly larger than its angle of line-of-action. Figure 2e illustrates the kinematical diagram

Machines 2021, 9, x FOR PEER REVIEW 5 of 21

of the 6PUS-2HKP mechanism with one PUS linkage, as the rest of them are the same. Figure 2f shows the composition of one PUS linkage. The 3D model of the RAPR and its corresponding prototype are shown in Figure 3. This RAPR is bio-inspired by human evolved masticatory system. In particular, the redundant actuation is achieved by adding passive kinematic constraints, the HKPs. The MachinesDOFs2021, 9of, 171 the robot reduce while the number of actuators remain the same. The passive5 of 22 point-contact HKP in the RAPR brings new challenges in stiffness analysis and driving force optimization, but it also helps to reproduce the chewing force more realistically and provide an in vitro6PUS-2HKP experimental mechanism platform with one to PUS study linkage, the as function the rest of them of TMJs are the insame. human Figure 2f shows the composition of one PUS linkage. masticatory system. Fossa structure TMJ structure Fossa surface Articular Fossa structure Condyle rod surface Mouth-closing position

Condyle Mouth-opening rod Condyle position Mandible structure (a) (b) (c) Temporalis linkage βT =14° Lateral pterygoid Prismatic Motor joint (P) linkage Ball screw Coupling Universal βL =50° β =30° M joint (U) Point contact P Masseter HKP linkage U Mandible S Singularity structure position Fixed Spherical joint (S) Fixed (d) (e) (f) FigureFigure 2. 2.Mechanism Mechanism of the of 6PUS-2HKP the 6PUS RAPR:-2HKP (a) RAPR: CT scan of(a the) CT fassa scan surface; of the (b) fassa design surface of the HKP;; (b ()c )design the layout of ofthe the two HKPs; (d) Lateral view of three linkages with symmetrical layout; (e) Kinematical diagram of the RAPR with one PUS HKP; (c) the layout of the two HKPs; (d) Lateral view of three linkages with symmetrical layout; linkage (Other linkages are omitted); (f) design of one PUS linkage (P: prismatic joint, U: Universal joint, S: Spherical joint). (e) Kinematical diagram of the RAPR with one PUS linkage (Other linkages are omitted); (f) design of one PUS linkage The(P: prismatic 3D model joint, of the U: RAPR Universal and its correspondingjoint, S: Spherical prototype joint) are. shown in Figure3 . This RAPR is bio-inspired by human evolved masticatory system. In particular, the redundant actuation is achieved by adding passive kinematic constraints, the HKPs. The Six motorsDOFs of the robot reduce while the number of actuators remain the same. The passive point-contact HKP in the RAPR brings new challenges in stiffness analysis and driving force optimization, but it also helps to reproduce the chewing force more realistically Two HKPs and provide an in vitro experimental platform to study the function of TMJs in human masticatory system.P U PUS linkage The left and right articular surface L and R (indicated in Figure4) are the surfaces S s s where the centers of the left and right condyles can move along. Each curved surface is made up of twoMandible sections, which can be described by two quadratic polynomial curves in XM-ZM plane as follows with respect to frame ΣL0 or ΣMa (see Section 2.3 Figure4).

 z = − x2 − x ( ≤ x < )  0.07 0.2 0 8 = ( ) 2 2 570 mm z f x (x − 12.7825) + (z + 2.4568) = 36 (8 ≤ x ≤ 13) (1)  − ≤ ≤ − ≤ ≤ (a) 3D model of the RAPR in SolidWorks 10 yL (10,b) Prototype130 yofR the110 RAPR Figure 3. Redundantlywhere actuatedx could chewing be xL or robotxR and. z could be zL or zR. They are coordinates of the center of two TMJs. The subscripts L and R represent the left and right TMJ center, respectively.

The left and right articular surface Ls and Rs (indicated in Figure 4) are the surfaces where the centers of the left and right condyles can move along. Each curved surface is made up of two sections, which can be described by two quadratic polynomial curves in XM-ZM plane as follows with respect to frame ΣL0 or ΣMa (see Section 2.3 Figure 4).

Machines 2021, 9, x FOR PEER REVIEW 5 of 21

of the 6PUS-2HKP mechanism with one PUS linkage, as the rest of them are the same. Figure 2f shows the composition of one PUS linkage. The 3D model of the RAPR and its corresponding prototype are shown in Figure 3. This RAPR is bio-inspired by human evolved masticatory system. In particular, the redundant actuation is achieved by adding passive kinematic constraints, the HKPs. The DOFs of the robot reduce while the number of actuators remain the same. The passive point-contact HKP in the RAPR brings new challenges in stiffness analysis and driving force optimization, but it also helps to reproduce the chewing force more realistically and provide an in vitro experimental platform to study the function of TMJs in human masticatory system. Fossa structure TMJ structure Fossa surface Articular Fossa structure Condyle rod surface Mouth-closing position

Six motors

 z 0.07 x2  0.2 x (0  x  8) Two HKPs  P z f() x (x 12.7825)2  ( z  2.4568) 2  36 (8  x  13) (1) U PUS linkage S  10 yLR  10,  130  y  110 Mandible where x could be xL or xR and z could be zL or zR. They are coordinates of the center of two TMJs. The subscripts L and R represent the left and right TMJ center, respectively.

5702.3. mm Reference Frames (a) 3D model of the RAPR in SolidWorks (b) Prototype of the RAPR Three reference frames are set up in order to describe the motion of the robot, as Figure 3. Redundantly actuatedFigure chewing 3. Redundantly robot. actuated chewing robot. shown in Figure 4 Bi (i = 1, …, 6) indicates the positions of six prismatic joints on the base. 2.3.Ci Referencerepresents Frames the connection point between the prismatic joint and universal joint. Mi The left and right articular surface Ls and Rs (indicated in Figure 4) are the surfaces representsThree reference the center frames arepoint set upof inthe order spherical to describe joint. the motionThe spherical of the robot, joints as are attached to the where the centers of shownthe left in and Figure right4 Bi (condylesi = 1, ... , 6) can indicates move the along positions. Each of six curved prismatic surface joints onis the mandibular platform. CiMi denotes the six driving rods which link the universal joint and made up of two sections,base. Cwhichi represents can be the described connection pointby two between quadratic the prismatic polynomial joint and curves universal in joint. M a M M M Ma XM-ZM plane as followsthei withrepresents spherical respect the centerjoint. to frame pointMandible Σ ofL0 the or spherical ΣframeMa (see joint.M Section-X TheY spherical 2.3Z Figure (Σ joints) is4). arefixed attached at the to home position of the theleft mandibular TMJ center platform. whenCiM thei denotes lower the and six driving upper rods teeth which are link in the occlusion. universal joint Global reference frame and the spherical joint. Mandible frame Ma-XMYMZM (ΣMa) is ﬁxed at the home position ofO theB-X leftBY TMJBZB center(ΣB) is when established the lower and on upper the base teeth arejust in under occlusion. ΣMa Global. The reference left instantaneous frame L- frameXLYLOZBL-X (BΣYLB),Z Bwhich(ΣB) is establishedmoves with on thethe base mandible, just under isΣMa placed. The left at instantaneous the center of left TMJ. Frame ΣB frameis locatedL-XLYL atZL Z(ΣML ),= which−167 movesmm, withhaving the mandible,the same is placedorientation at the center with of Σ leftMa TMJ.. Point L and R represent Frame ΣB is located at ZM = −167 mm, having the same orientation with ΣMa. Point L and Rthe represent left and the leftright and center right center point point of of two two HKPs,HKPs, respectively. respectively.

D ZM YM R0 ZR Ma(L0) ZL YL Ci YR L XM Plane P1 R XR XL Right curved surface Rs Left curved Plane P2 Mi surface Ls Qi Incisor point I ZB YB Trajectory

OB XB Front

Figure 4. Reference frames and schematic diagram of the robot (Other ﬁve linkages are omitted in thisFigure ﬁgure). 4. Reference frames and schematic diagram of the robot (Other five linkages are omitted in this figure). The pose of end-effector with respect to the base can be described by three arbitrary non-collinear points. The three points may be selected anywhere on the end-effector. Generally,The the pose platform of end joints-effector of a PR are with in a same respect plane. to While, the thebase redundantly can be described actuated by three arbitrary non-collinear points. The three points may be selected anywhere on the end-effector. Generally, the platform joints of a PR are in a same plane. While, the redundantly actuated chewing robot is a 3D spatial one and all spherical joints Mi and universal joints Ci of this robot are not in the same plane. The expression of coordinates of platform joints in global frame in terms of three end-effector points will be more complicated than the robot whose platform joints are in same plane.

3. Differential Kinematics 3.1. Inverse Kinematics The three end-effector points are selected as L, R and I, as shown in Figure 4. The coordinates of platform spherical joints in mandible frame in terms of three end-effector points can be expressed as:

LMLRLILDik1, i  k 2, i  k 3, i (2)

Machines 2021, 9, 171 7 of 22

chewing robot is a 3D spatial one and all spherical joints Mi and universal joints Ci of this robot are not in the same plane. The expression of coordinates of platform joints in global frame in terms of three end-effector points will be more complicated than the robot whose platform joints are in same plane.

3. Differential Kinematics 3.1. Inverse Kinematics The three end-effector points are selected as L, R and I, as shown in Figure4. The coordinates of platform spherical joints in mandible frame in terms of three end-effector points can be expressed as:

LMi = k1,iLR + k2,iLI + k3,iLD (2)

where LD = LR × LI. Vector LD is perpendicular to both LR and LI. Then, for each i, the three unknowns (k1,i, k2,i and k3,i) in Equation (2) can be obtained in terms of initial position coordinates of points L, R, I and Mi. In consideration of the HKPs constrains, four generalized variables are able to describe the position and pose of the robot. Assuming G is the generalized variables, four independent parameters xI, yI, zI and xL are chosen to describe the motion of the robot. The rest of three end-effector points’ coordinates yL, zL, xR, yR and zR can be calculated based on the geometric constraints. These nine coordinate parameters of three end-effector points are deﬁned as E. Left HKP keeps contact with constrained surface, we have:

zL = f (xL) (3)

where, f indicates the constraint Equation (1). zL can be calculated. The length of LI is constant, then we have: kLIk = lLI (4)

From Equation (4), yL can be obtained. Similarly, as RL and RI all have ﬁxed length and right HKP keeps contact with right constrained surface, the following equations can be written:  ( − )2 + ( − )2 + ( − )2 = k k2  xR xL yR yL zR zL RL  2 2 2 2 (xR − xI) + (yR − yI) + (zR − zI) = kRIk (5)   zR = f (xR)

where xR, yR and zR can also be obtained by using Equation (5). Then the coordinates of platform joints Mi can be calculated from Equation (2). Based on the geometric constraints of parallel kinematic chains, the joint variables Qi(i=1, 2, . . . , 6) of the 6 PUS kinematic chains can be obtained [46].

3.2. Velocity Analysis and Jacobian Matrix Jacobian matrix is the generalized transmission ratio of the velocity or force transmission from the input of the joint variables to the output of the generalized variables [47]. This RAPR has mixed translational and rotational DOFs. For this type of RAPR, this section presents a dimensionally homogeneous non-square Jacobian matrix formulation method based on three end-effector points. For the robot studied in this paper, the length of leg CiMi is constant and leads to:

2 2 kCiMik = Li (6)

Differentiating (6) yields: · = · VMi CiMi VCi CiMi (7) Machines 2021, 9, 171 8 of 22

where VM is the velocity of spherical joint, and VC the velocity of prismatic joint, indicated . i i by Qi. The position of Mi in global frame can be expressed as:

OBMi = OBL + k1,iLR + k2,iLI + k3,iLD (8)

Differentiating (8) gives:

. . . . . = = − − ) + + + [ × ( − ) VMi Mi (1 k1,i k2,i L k1,iR k2,iI k3,i R I L . . (9) +L × (R − I) + I × (L − R)]

where I (short for OBI) is coordinate of point I in global frame, andL and R are the . . . . iT . . . . T . . . . T abbreviations in the same way. L= [xL, yL, zL , R = xR, yR, zR and I = xI, yI, zI . Combining (7) and (9), yields:  .   A A A  .  e · C M 0 ··· 0  Q 1,1 2,1 3,1   1 1 . 1 L · ···    A1,2 A2,2 A3,2  .  0 e C2M2 0   Q2    ·   =   · (10)  . . .   R   . . . .   .   . . .  .  . . . .   .  I  .  A A2,6 A3,6 0 0 ··· e · C6M6 1,6 Q6

T where A1,i = [(1 − k1,i − k2,i) · CiMi + k3,i(R − I) × CiMi] , T A2,i = [k1,iCiMi + k3,i(I − L) × CiMi] , and T A3,i = [k2,iCiMi + k3,i(L − R) × CiMi] , i = 1, 2, . . . , 6. . . . If nine velocity variables L, R and I are known, the velocities of prismatic joints . . . . Q (Q , Q , ··· , Q ) can be calculated by (10). While, we only know four generalized i 1 2 6 . . . . velocity variables xI, yI, zI, xL. The ﬁve remaining velocity variables can be obtained by velocity projection theory. On a rigid body, projection of velocity vectors of any two points on the line through these two points are the same. For example, regarding velocities of point I and L, we have: ...... [xI, yI, zI] · LI = [xL, yL, zL] · LI (11) T in which, LI = [xLI, yLI, zLI] . The instantaneous velocity vector of a condyle point is in a plane that goes through the condyle center and is tangent to the curved surface as shown in Figure4, Plane P 1 and P2. The slope of P1 in Xm-Zm plane can be calculated by the equation of left curved surface Ls. So does the slope of P2: = 0 ( ) mP1 f L xL (12)

Since the resultant velocity vector VL must be in the plane P1, the slope mP1 can also be calculated by: . . = · zL mP1 xL (13) . . . . . Substituting zL into (11), yL can be obtained. Similarly, xR, yR, zR can also be obtained. Writing in matrix form, we have:

. . E(9×1) = P(9×4) · G(4×1) (14)

where P is a matrix that can be obtained by velocity projection theory aforementioned, and . h . . . iT E(9×1) = L, R, I . Equation (10) can be rewritten as:

. . A(6×9) · E(9×1) = B(6×6) · Q(6×1) (15) Machines 2021, 9, 171 9 of 22

Combining (14) and (15), gives:

. . Q = B−1 · A · P · G (16) (6×1) (6×6) (6×9) (9×4) (4×1)

The inverse Jacobian matrix of the RAPR can be written as:

−1 −1 JRA = B · A · P (17)

−1 JRA is a 6 × 4 matrix.

4. Stiffness Analysis and Optimization of Driving Force 4.1. Stiffness Analysis Stiffness is a very important performance index in parallel robots [48]. Redundant actuation has been proved to be an effective way to improve its stiffness [26]. The redundant actuation of this 6PUS-2HKP RAPR is achieved by adding two passive point-contact constraints HKPs to the 6-PUS parallel mechanism. In order to investigate the effect of point- contact HKP constraint on the redundantly actuated chewing robot, the stiffness analysis of redundantly actuated 6PUS-2HKP and non-redundantly actuated 6-PUS mechanism are compared. The relation between Cartesian generalized velocity and joint velocity can be obtained by combining Equations (16) and (17):

. . −1 Q = JRA · G (18)

The inﬁnitesimal displacements in the Cartesian space and joint space can be written as:

−1 ∆Q = JRA · ∆G (19)

Based on virtual work principle, the relation between external force F and prismatic joint force τ can be written as: T T τ · ∆Q = FG · ∆G (20) The joint force τ can also be computed by:

τ = KC · ∆Q (21)

where KC is a diagonal matrix which consists of stiffness of the driving system. Combining Equations (19)–(21), the following equation can be obtained.

−T −T −1 FG = JRA · KC · ∆Q = JRA · KC · JRA · ∆G (22)

Therefore, the stiffness matrix of the RAPR can be deﬁned as:

−T −1 K = JRA · KC · JRA (23)

The generalized inﬁnitesimal displacements of the end-effector can be expressed by:

−1 ∆G = K · FG (24)

In this derivation, the basic assumptions in this paper are: (1) The end-effector and the base are assumed as rigid bodies, as the rigidities of them are much larger than that of the six driving rods; (2) the weights of the legs are neglected; (3) the joints are assumed rigid and frictionless; (4) the stiffness of the motors and coupler is equal and is assumed as rigid. All of the six motors actuate the six prismatic joints through ball screw and they have the same rigidity. In one PUS linkage, only the ball screw and the driving legs are taken into account. As the prismatic joints are the direct actuated joints, the model can be Machines 2021, 9, x FOR PEER REVIEW 10 of 21 Machines 2021, 9, x FOR PEER REVIEW 10 of 21 Machines 2021, 9, 171 10 of 22

InIn th the e numerical numerical simulation, simulation, a amouth mouth-opening-opening trajectory trajectory was was implemented implemented to to simplified as follows. The stiffness matrix of one linkage is approximately calculated by estimateestimate the the deformation deformation displacement displacement occurring occurring to to 6PUS 6PUS-2HKP-2HKP RAPR RAPR and and 6 -6PUS-PUS PR. PR. theThe followingmotion of equation: the RAPR in terms of the incisor point (point I) with respect to frame ΣB is The motion of the RAPR in terms of the−1 incisor− point1 (point−1 I) with respect to frame ΣB is shown in Figure 5. The motion lastedKCi =1 sK andbsi a +resultanKdri t force (80 N in ZB-direction (25)and shown in Figure 5. The motion lasted 1 s and a resultant force (80 N in ZB-direction and 20 N in XB-direction) was continuously applied to point I. During this process, the 20where N inK XCBis-direction) the scalar wasstiffness continuously parameter. appliedKbs and K todr pointstand I for. During the stiffness this process,parameters the of deformation displacements G at different configuration of the robot were calculated, deformationthe ball screw displacements and driving rod,G respectively. at different Kconfigurationbs and Kdr can of bethe computed robot were as: calculated, as shown in Figure 6. The largest deformation displacement is 4.047 × 10−6 m in ZI direction as shown in Figure 6. The largest deformation displacement is 4.047 × 10−6 m in ZI direction and the smallest one is 2.573 × 10−7 m Kin =-YIE direction.· A /L It can be seen that the RAPR at each(26) and the smallest one is 2.573 × 10−7 m in -jYI direction.j j jIt can be seen that the RAPR at each configurationconfiguration has has small small infinitesimal infinitesimal displacements, displacements, which which means means that that it it owns owns a a high high where E is the Young’s modulus of the material, A the sectional area of the link, L the length of stiffness.stiffness. the link,To and highlightj = bs, dr. the The role scalar of stiffness redundant parameters actuation of six in linkages improving can computed the stiffness, as follows. the To highlight the 7 role of redundant actuation7 in improving the stiffness,7 the KC1 = KC2 = 2.11 × 10 N/m, KC3 = KC4 = 1.88 × 10 N/m, KC5 = KC6 = 2.57 × 10 N/m. deformationdeformation displacement displacement of of the the non non-redundantly-redundantly actuated actuated 6- 6PUS-PUS PR PR was was also also computed. computed. In the numerical simulation, a mouth-opening trajectory was implemented to estimate TheThe displacement displacement of of the the 6- 6PUS-PUS PR PR under under the the same same mouth mouth-opening-opening trajectory trajectory is isshown shown in in theFigure deformation 7. Having displacement the same geometric occurring parameters to 6PUS-2HKP with the RAPR 6PUS and-2HKP 6-PUS RAPR PR., the The smallest motion Figureof the 7. RAPR Having in the terms same of geometric the incisor parameters point (point withI) withthe 6PUS respect-2HKP to frame RAPRΣ, theis smallest shown in deformation displacement of the non-redundantly actuated 6-PUS PR is 1.451B × 10−5 in XI deformation displacement of the non-redundantly actuated 6-PUS PR is 1.451 × 10−5 in XI Figure5 . The motion lasted 1 s and a resultant force (80 N in ZB-direction and 20 N in direction,direction, which which is is larger larger than than any any deformation deformation displacements displacements of of the the RAPR RAPR at at each each XB-direction) was continuously applied to point I. During this process, the deformation configuratconfiguration.ion. By By comparison comparison of of these these two two robots, robots, one one can can see see that that the the redundantly redundantly displacementsactuated 6PUS-∆2HKPG at different RAPR experiences configuration smaller of the deformation robot were displacements calculated, as undershown the in actuated 6PUS-2HKP RAPR experiences smaller deformation× displacements−6 under the Figuresame force.6. The It can largest be concluded deformation from displacement the numerical is simulation 4.047 10 that mthe in 6PUSZI direction-2HKP RAPR and samethe force. smallest It can one be is concluded 2.573 × 10 from−7 mthe in numerical -Y direction. simulation It can that be the seen 6PUS that-2HKP the RAPR RAPR at with two HKPs mimicking human TMJs ownsI better stiffness than the non-redundantly witheach two configuration HKPs mimicking has small human infinitesimal TMJs owns displacements, better stiffness which than the means non- thatredundantly it owns a actuated 6-PUS PR counterpart. actuatedhigh stiffness. 6-PUS PR counterpart.

Figure 5. Movement of point I with respect to frame ΣB. FigureFigure 5. 5.MovementMovement of ofpoint point I with I with respect respect to toframe frame ΣBΣ. B.

Figure 6. Deformation displacement of 6PUS-2HKP RAPR. Figure 6. DeformationDeformation displacement displacement of of6PUS 6PUS-2HKP-2HKP RAPR RAPR..

Machines 2021, 9, 171 11 of 22

To highlight the role of redundant actuation in improving the stiffness, the deformation displacement of the non-redundantly actuated 6-PUS PR was also computed. The displacement of the 6-PUS PR under the same mouth-opening trajectory is shown in Figure7 . Having the same geometric parameters with the 6PUS-2HKP RAPR, the smallest −5 deformation displacement of the non-redundantly actuated 6-PUS PR is 1.451 × 10 in XI direction, which is larger than any deformation displacements of the RAPR at each con- ﬁguration. By comparison of these two robots, one can see that the redundantly actuated 6PUS-2HKP RAPR experiences smaller deformation displacements under the same force. Machines 2021, 9, x FOR PEER REVIEW 11 of 21 It can be concluded from the numerical simulation that the 6PUS-2HKP RAPR with two HKPs mimicking human TMJs owns better stiffness than the non-redundantly actuated 6-PUS PR counterpart.

FigureFigure 7.7.Deformation Deformation displacement displacement of non-redundantly of non-redundantly actuated 6-PUS actuated PR. 6-PUS PR. 4.2. Optimization of Driving Force 4.2. OptimizationThe inclusion of of HKP Driving into 6-PUS Force mechanism brings new features to the chewing robot. On theThe one inclusion hand, redundant of HKP actuation into 6of-PUS the RAPR mechanism causes the brings inverse dynamicsnew features of robot to the chewing robot.lacks a On unique the solution. one hand, On theredundant other hand, actuation it also allows of redistributionthe RAPR causes of driving the force inverse by dynamics of dynamics. This section will investigate the effect of point-contact HKP constraint in this robotredundant lacks actuation a unique system solution. from the On perspective the other of hand, dynamics. it also The allows driving forcesredistribution of the of driving forceRAPR by are dynamics. optimized by This finding section the minimum will investigate Euclidean normthe effect solution. of point The optimization-contact HKP constraint inflow this chart redundant is shown inactuation Figure8. The system inverse from kinematics the perspective has been discussed of dynamics. in Section The 3.1 .driving forces ofFrom the this RAPR foundation, are optimized the velocity, kinetic by finding energy, potential the minimum energy of Euclideanthe six PUS linkages norm solution. The optimizationand end-effector flow can chart be calculated, is shown the in details Figure of 8 which. The areinverse not provided kinematics due tohas the been discussed limited space. in SectionLagrange’s 3.1. equationsFrom this are foundation, used to derive the the velocity, dynamics ofkinetic the RAPR. energy, The Lagrange’spotential energy of the sixequations PUS linkages of the first and type end can be-effector written incan two be sets. calculated The first equation, the details set can of be which written are as: not provided due to the limited space. 6 ∂Γi d ∂L ∂L ∑ λi = ( . ) − − Qgj (j = 1, . . . , k) (27) i=1 ∂qj dt ∂qj ∂qj Planned trajectory of robot, including whereInput L is the Lagrange function;positionQgj ,represents velocity, and the a generalizedcceleration force on the end-effector and k is the number of generalized coordinates, k = 4; i indicates the number of constraint equations, i = 6; n is the sum of i and k, n = 10; Γi is the ith constraint equation; qj indicates the four generalized positionConstraints parameters of six XL, YL, βLConstraintsand γL in Cartesianof space. βL and T γL are the Euler angles.PUS Let GlinkagesC be [xL, yL, βL, γL] .two As HKPs the robot has four DOFs, the T generalized forces Qgj = [Fgx, Fgy, Mgβ, Mgγ] are defined as forces in XL and YL direction

Kinematics Inverse kinematics

Calculation of velocity, kinetic energy, potential energy of each components in the robot

Dynamics The first equation set Lagrange’s equations of the first type The second equation set

The pseudo-inverse Optimization method method

Output Six minimum 2-norm driving forces

Figure 8. The flow chart of the driving force optimization.

Lagrange’s equations are used to derive the dynamics of the RAPR. The Lagrange’s equations of the first type can be written in two sets. The first equation set can be written as:

Machines 2021, 9, x FOR PEER REVIEW 11 of 21

Machines 2021, 9, 171 12 of 22

and torques around YL and ZL. Considering the geometric constraints of each kinematic chain, link CiMi has a ﬁxed length Li and leads to:

2 2 Γi = kCiMik − Li = 0 (i = 1, 2, . . . , 6) (28)

The second equation set, relating the driving torque/force (driving force of slider in this paper), can be expressed by:

k d ∂L ∂L ∂Γi τi = Qaj = ( . ) − − ∑ λi (j = k + 1, . . . , n) (29) Figure 7. Deformation displacementdt ∂qj ∂q ofj noni=1 -redundantly∂qj actuated 6-PUS PR.

4.2.where Optimizationτi (Qaj) is the of driving Driving force. Force According to the optimization algorithm of pseudo-inverse method, the minimum 2-normThe solution inclusionτ0 can of be obtained:HKP into 6-PUS mechanism brings new features to the chewing

robot. On the one hand, redundantRM actuation of the RAPR causes the inverse dynamics of 0 ∂Γ ∂Γ −1 ∂Γ ∂Γ −1 d ∂L ∂L τ = [ · ( ) ] · [ · ( ) ] · [ ( . ) − ] robot lacks a unique∂GC solution.∂Q On∂ GtheC other∂Q hand,dt it also ∂allowsQ redistribution of driving 6×4 4×6 ∂Q 6×1 (30) force by dynamics. This sectionRM will investigate the effect of point-contact HKP constraint ∂Γ ∂Γ −1 d ∂L ∂L −[ · ( ) ] · [ ( . ) − − Fg] ∂GC ∂Q × dt ∂GC in this redundant actuation6 4 system∂GC from the perspective4×1 of dynamics. The driving forces of the RAPR are optimized by finding the minimum Euclidean norm solution. The −1 RM −1 where [ ∂Γ · ( ∂Γ ) ] is the right pseudo-inverse matrix of [ ∂Γ · ( ∂Γ ) ] . τ0 denotes optimization∂GC ∂Q flow6× chart4 is shown in Figure 8. The inverse∂GC ∂Q kinematics4×6 has been discussed inthe Section six minimum 3.1. From 2-norm this driving foundation, forces. the velocity, kinetic energy, potential energy of the six PUSThe trajectorylinkages ofand left end lateral-effector movement can wasbe calculated analyzed in, the this details simulation, of which shown are in not provided Figure9. The position, velocity, and acceleration of the six prismatic joints for the given dposeue to of mandiblethe limited can bespace. determined by inverse kinematics.

Planned trajectory of robot, including Input position, velocity, and acceleration

Constraints of six Constraints of PUS linkages two HKPs

Kinematics Inverse kinematics

Calculation of velocity, kinetic energy, potential energy of each components in the robot

Dynamics The first equation set Lagrange’s equations of the first type The second equation set

The pseudo-inverse Optimization method method

Output Six minimum 2-norm driving forces

Figure 8. The ﬂow chart of the driving force optimization. Figure 8. The flow chart of the driving force optimization.

Lagrange’s equations are used to derive the dynamics of the RAPR. The Lagrange’s equations of the first type can be written in two sets. The first equation set can be written as:

Machines 2021, 9, x FOR PEER REVIEW 12 of 21

6  d L  L i  i( )  Q gj ( j  1, , k ) (27) i 1 qj dt  q j  q j

where L is the Lagrange function; Qgj represents the generalized force on the end-effector and k is the number of generalized coordinates, k = 4; i indicates the number of constraint equations, i = 6; n is the sum of i and k, n = 10; Γi is the ith constraint equation; qj indicates the four generalized position parameters XL, YL, βL and γL in Cartesian space. βL and γL are the Euler angles. Let GC be [xL, yL, βL, γL]T. As the robot has four DOFs, the generalized T L L forces QFFMMgj  [,,,] gx gy g β gγ are defined as forces in X and Y direction and torques around YL and ZL. Considering the geometric constraints of each kinematic chain, link CiMi has a fixed length Li and leads to:

2 2 i CM i i L i 0 ( i  1,2,...,6) (28)

The second equation set, relating the driving torque/force (driving force of slider in this paper), can be expressed by:

k d L  L  i iQ aj ( )    i ( j  k  1, , n ) (29) dt q j  q ji 1  q j

where τi (Qaj) is the driving force. According to the optimization algorithm of pseudo-inverse method, the minimum 2- norm solution τ0 can be obtained: Γ Γ Γ Γ d  L  L τ 0[()][()][()]  1 RM    1   6 4 4  6 6  1 GGCC QQQ  dt Q  (30) Γ Γ d  L  L [()][] 1 RM （）-  F 64 g 4  1 GGCC Q dt GC 

Γ Γ Γ Γ where [()] 1 RM is the right pseudo-inverse matrix of [()] 1 . τ0 denotes Machines 2021, 9, x FOR PEER REVIEW G Q 6 4 G Q 4 613 of 21 C C the six minimum 2-norm driving forces. The trajectory of left lateral movement was analyzed in this simulation, shown in

Machines 2021, 9, 171 Figure 9. The position, velocity, and acceleration of the six prismatic joints for the13 given of 22

) 2

. s 150 10 / pose of mandible can α.beL determined by inverse kinematics.

) γL

deg

/ (

100

5 deg

( 3 4.5

Position αL

Velocity 50 4 γL 0 2 ) 3.5 ) s deg / 0 ( 3 1 2.5 –5 mm ..

) ( 0 –50 2 α..L

1.5 γL Euler angle Euler rate mm

–10 ( –1

–100 angle Euler

L 1 Euler angle Euler acceleration 0 0.2 0.4 Y 0–.26 0.8 1 1.2 1.4 0 0.2 0 0.5.4 0.6 0.8 1 1.2 1.4 Time (s) Time (s) 0 Machines 2021, 9, x FOR PEER REVIEW –3 0 0.2 0.4 0.6 0.8 1 1.2 131.4 of 21 (c) Euler angle0 0rate.2 0.4 , 0. 6 0.8 1 1.2 1.4 (d) Euler angle acceleration , LL Time (s) LL Time (s) Figure 9. Input trajectory(a) Position parameters and velocityof the RAPR of input. YL (b) Euler angle αL and γL ) 2

. s 150 10 / α. L ) γL s deg

Following the same/ lateral movement, the driving force and driving power of 6PUS- (

100 5 deg

2HKP RAPR and its( counterpart non-redundantly actuated 6-PUS parallel robot were 50 solved by dynamics0 equations for comparison. The driving force and power of 6-PUS robot without optimization algorithm are illustrated in Figure0 10 and Table 1. The driving –5 .. forces from the most significant to the least significant in sequence–50 are masseter, lateralα..L γL pterygoid, and temporalis. angleEuler rate –10 The maximum force is that of –LM100 which is 10.78 N and the minimum force is that 0of RT0.2 which 0.4 0.6 is0.8 6.27 1 N. 1.2 As 1.4 the velocityangleEuler acceleration of0 RLP 0.2 is0.4 the 0.6 largest, 0.8 1 the 1.2 RLP 1.4 Time (s) Time (s) has a maximum power of 0.487 W. (c) Euler angle rate LL,   (d) Euler angle acceleration LL,  The driving forces of the 6PUS-2HKP RAPR adopting pseudo-inverse optimization method are shownFigureFigure in9. InputFigure 9. Input trajectory 1 trajectory1 and parameters Table parameters 1 .of Compared the of the RAPR RAPR.. with the 6-PUS robot, the driving forces of the six prismatic joints reduces dramatically, especially the lateral pterygoid and FollowingFollowing the thesame same lateral lateral movement, movement, the the driving driving force force and and driving driving power power of 6PUS-6PUS- the masseter. The2HKP 2HKPmaximum RAPR RAPR and force and its counterpartits iscounterpart that of RM non non-redundantly -whichredundantly is 10.26 actuated actuated N and 6 6-PUSthe-PUS minimum paralle parallell robot force were were solved by dynamics equations for comparison. The driving force and power of 6-PUS robot occurs at the RLPsolved which by dynamics has a value equations of 3.36 for N. comparison. As we can seeThe fromdriving the force standard and power deviation of 6-PUS without optimization algorithm are illustrated in Figure 10 and Table1. The driving forces of the power inrobot Table without 1, the optimization driving power algorithm also are becomes illustrated more in Figure balanced 10 and and Table the 1. Thepower driving from the most signiﬁcant to the least signiﬁcant in sequence are masseter, lateral pterygoid, consumption forces has and improved. from temporalis. the most The The significant maximum RT has to force a the maximum isleast that ofsignificant LM power which in is 0.224sequence 10.78 N W. and are None the masseter, minimum of any lateral force kinematic chainpterygoid, ofis 6PUS that ofand-2HKP RT temporalis. which RAPR is 6.27 Thehas N. As maximuma thevery velocity sharp force of powe RLPis that isr the curveof largest,LM comparedwhich the RLP is 10.78 has with a N maximum andthe the 6-PUS robot. minimumpower force of 0.487 is that W. of RT which is 6.27 N. As the velocity of RLP is the largest, the RLP has a maximum power of 0.487 W. The driving forces of the 6PUS-2HKP RAPR adopting pseudo-inverse optimization 11 RLP method are shown in Figure 11 and Table500 1. Compared with the 6-PUS robot,RLP the driving

LLP ) ) 10 400 LLP N forces of the six prismaticRM joints reduces dramatically, especially the lateral pterygoid and ( RM

LM mW 300 9 the masseter. The maximum force is ( that of RM which is 10.26 N and the LMminimum force RT 200 RT 8 occurs at the RLP which hasLT a value of 3.36 N. As we can see from the standard deviation 100 LT 7 of the power in Table 1, the driving power0 also becomes more balanced and the power 6 consumption has improved. The RT– 100 has a maximum power 0.224 W. None of any kinematic chain of 6PUS-2HKP RAPR– 200has a very sharp power curve compared with the

PUS driving force force PUSdriving 5

- 6-PUS robot. –300

6 PUSpower driving 4 - 6 –400 3 11 RLP–500 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 500 0.4 0.6 0.8 1 1.2 1.4 1.6 RLP

LLP ) ) 10 400 LLP N Time (s) RM Time (s) ( RM

LM mW 300 9 ( LM (a) Driving force RT (200b) Driving power RT 8 LT 100 LT FigureFigure 10. Driving 10. Driving force and force power7 and of power non-redundant of non 6-PUS-redundant parallel 6 mechanism-PUS parallel0 (Right mechani lateral pterygoid-RLP;sm (Right lateral left lateral pterygoid-LLP;pterygoid- rightRLP; masseter-RM; left lateral6 leftpterygoid masseter-LM;-LLP; right right temporalis-RT; masseter-RM; left– temporalis-LT).100left masseter-LM; right temporalis- –200

RT; left temporalis PUS force driving -LT)5 . - –300 6 PUS driving power power PUS driving 4 -

6 –400 3 –500 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Time (s) Time (s) (a) Driving force (b) Driving power Figure 10. Driving force and power of non-redundant 6-PUS parallel mechanism (Right lateral pterygoid-RLP; left lateral pterygoid-LLP; right masseter-RM; left masseter-LM; right temporalis- RT; left temporalis-LT).

Machines 2021, 9, 171 14 of 22

Table 1. Comparison of driving force and power between 6-PUS and 6PUS-2HKP mechanism (Fmin, Fmax, Fmean-The minimum, maximum and mean value of the driving force, respectively; Pmax-The maximum value of the power; PSD-The standard deviation of the power).

Mechanism Parameters RLP LLP RM LM RT LT

Fmin (N) 7.71 8.25 9.60 10.49 6.27 6.48 Fmax (N) 8.48 8.65 10.41 10.78 6.71 6.65 6-PUS Fmean (N) 8.18 8.53 10.04 10.61 6.56 6.54 Pmax (mW) 487.4 103.3 152.9 68.2 226.3 70.7 PSD 340.3 70.7 100.0 44.8 154.4 48.3

Fmin (N) 3.36 6.04 7.53 7.39 6.25 6.46 Fmax (N) 7.71 7.47 10.26 8.67 6.68 6.63 6PUS-2HKP Fmean (N) 4.62 6.42 8.45 7.76 6.55 6.53 Pmax (mW) 216.3 73.9 117.9 47.8 224.4 70.6 PSD 157.9 50.8 77.7 31.5 154.1 48.2

The driving forces of the 6PUS-2HKP RAPR adopting pseudo-inverse optimization method are shown in Figure 11 and Table1. Compared with the 6-PUS robot, the driving forces of the six prismatic joints reduces dramatically, especially the lateral pterygoid and the masseter. The maximum force is that of RM which is 10.26 N and the minimum force occurs at the RLP which has a value of 3.36 N. As we can see from the standard deviation Machines 2021, 9, x FOR PEER REVIEW 14 of 21 of the power in Table1, the driving power also becomes more balanced and the power consumption has improved. The RT has a maximum power 0.224 W. None of any kinematic chain of 6PUS-2HKP RAPR has a very sharp power curve compared with the 6-PUS robot.

11 RLP )

) RLP LLP 500 N LLP ( 10 mW 400 RM ( RM 9 LM 300 LM 8 RT 200 RT LT 100 LT 7 0 6 –100 5 HKP driving force HKPdriving force –200 2 HKP driving power driving HKP -

4 2 –300 - –400 PUS 3 6 PUS –500 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Time (s) Time (s) (a) Driving force (b) Driving power Figure 11. Driving force andFigure power 11. Driving of 6PUS force- and2HKP power RAPR of 6PUS-2HKP. RAPR. The highlight of the role of the point-contact HKP constraints in the 6PUS-2HKP Table 1. Comparison ofRAPR driving is homogenizing force and power driving between force and 6 driving-PUS and power. 6PUS The-2HKP driving mechanism force decreases (F andmin, Fmax, Fmean-The minimum,the drivingmaximum power and becomes mean more value homogenous of the driving by applying force, the respectively; pseudo-inverse P optimiza-max-The maximum value of the tionpower; method. PSD-The standard deviation of the power).

Mechanism Parameters5. Chewing ExperimentRLP LLP RM LM RT LT 5.1. Measurement Methods for Chewing Simulated Food Fmin (N)In this paper,7.71 the reaction8.25 forces on the9.60 occlusal surface10.49 of incisor6.27 and on the mandibu-6.48 Fmaxlar (N) condyle are8.48 regarded as the8.65 occlusal forces10.41 and the TMJ10.78 (HKP) forces,6.71 respectively. 6.65 For the purpose of studying the relation between occlusal force and TMJ force of the chewing 6-PUS Fmean (N) 8.18 8.53 10.04 10.61 6.56 6.54 robot, a real-time force measurement system was built. Due to the size of the point-contact Pmax HKP(mW) constraints, 487.4 the strain103.3 gauge is selected152.9 as the force68.2 sensor. Two226.3 strain gauges70.7 are PgluedSD perpendicular340.3 to each other70.7 on the100.0 upper jaw, as44.8 shown in Figure154.4 12 a. The48.3 vertical Fmin (N) 3.36 6.04 7.53 7.39 6.25 6.46 Fmax (N) 7.71 7.47 10.26 8.67 6.68 6.63 6PUS-2HKP Fmean (N) 4.62 6.42 8.45 7.76 6.55 6.53 Pmax (mW) 216.3 73.9 117.9 47.8 224.4 70.6 PSD 157.9 50.8 77.7 31.5 154.1 48.2

The highlight of the role of the point-contact HKP constraints in the 6PUS-2HKP RAPR is homogenizing driving force and driving power. The driving force decreases and the driving power becomes more homogenous by applying the pseudo-inverse optimization method.

5. Chewing Experiment 5.1. Measurement Methods for Chewing Simulated Food In this paper, the reaction forces on the occlusal surface of incisor and on the mandibular condyle are regarded as the occlusal forces and the TMJ (HKP) forces, respectively. For the purpose of studying the relation between occlusal force and TMJ force of the chewing robot, a real-time force measurement system was built. Due to the size of the point-contact HKP constraints, the strain gauge is selected as the force sensor. Two strain gauges are glued perpendicular to each other on the upper jaw, as shown in Figure 12a. The vertical one is used for measuring the occlusal force and the lateral one is for lateral force on teeth. The other two strain gauges are glued on left condylar rod and right condylar rod, respectively, as shown in Figure 12b. These two strain gauges sense the HKP force in real time.

Machines 2021, 9, 171 15 of 22

Machines 2021, 9, x FOR PEER REVIEW 15 of 21

one is used for measuring the occlusal force and the lateral one is for lateral force on teeth. The other two strain gauges are glued on left condylar rod and right condylar rod, Machines 2021, 9, x FOR PEER REVIEW Cross section of the middle 15 of 21 respectively, as shown in Figureof the strain12b. gauge These position two strain gauges sense the HKP force in real time. Vertical strain gauge Cross section of the middle Vertical strain gauge Upperof jaw the strain gauge position Upper jaw Vertical strain gauge

Vertical strain Lower jaw Lowergauge jaw Upper jaw Lateral strain gauge Lateral strain gauge Upper jaw (a) Fossa block Lower jaw Lower jaw Fossa block Condylar block Lateral strain gauge Lateral strain gauge Strain Cross section of the gauge middle of the strain (a) gauge position Lower jaw Fossa block Condylar block Strain gauge Pasted on both TMJs Fossa block Condylar block (b) Strain Cross section of the Figuregauge 12. Position of middlethe strain of the gauge: strain ( a) Strain gauge on upper jaw, (b) strain gauge on TMJ (HKP). gauge position Lower jaw Condylar block Strain gauge As the chewing robot was intended to performPasted on a both variety TMJs of trajectories between lateral and vertical chewing, the trajectories may vary. A desired chewing trajectory was profiled according to characteristics of human chewing trajectory. The given trajectories (b) of the mandible are determined by the four independent position parameters. The motion controllerFigureFigure 12.12. computesPosition Position of the of the thekinematics strain strain gauge: gauge: in (a )a Straintime (a) gaugeStraininterval on gauge of upper 2 ms on jaw, toupper (bget) strain the jaw, parameters gauge (b) strain on TMJ ofgauge (HKP). the on TMJ six(HKP). servo motors. The chewing trajectory of the chewing robot is shown in Figure 13 with respectAs to frame the chewing ΣB. robot was intended to perform a variety of trajectories between lateral and vertical chewing, the trajectories may vary. A desired chewing trajectory was SimulatedAs the food chewing is adopted robot to wasconduct intended chewing to tes performt on the chewing a variety robot of in trajectories order to between obtainproﬁled the according occlusal to force characteristics and TMJ of force. human Two chewing kinds trajectory. of materials The givensilicone trajectories (SI) and of lateral and vertical chewing, the trajectories may vary. A desired chewing trajectory was polyethylenethe mandible (PE) are are determined chosen as bythe the simulated four independent food, as shown position in Figure parameters. 14. As The different motion thicknessesprofiledcontroller according of computes simulated theto food kinematicscharacteristics can produce in a time differentof human interval value ofchewing 2 of ms force, to gettrajectory. silicone the parameters with The 4 mm, given of the8 trajectories mmofsix theand servo mandible 12 motors.mm thick Theare and determined chewing polyethylene trajectory by withthe of fourthe1 mm, chewing independent 2 mm robot thick is are position shown adopted in parameters. Figure during 13 thewith The motion chewingcontrollerrespect cycle to frame computes in thisΣB .paper. the kinematics in a time interval of 2 ms to get the parameters of the six servo motors. The chewing trajectory of the chewing robot is shown in Figure 13 with 128 128 Occlusal phase respect to frame ΣB. 126 126Simulated food is adoptedTeeth to conduct chewing test on the chewing robot in order to obtain124 the occlusal force 124 and TMJ force. Two kinds of materials silicone (SI) and Closing phase polyethylene122 (PE) are chosen122 as the simulated food, as shown in Figure 14. As different mm mm / / Z Z thicknesses Z 120 of simulated food120 can produce different value of force, silicone with 4 mm, 8 Openning mm118 and 12 mm thick and 118polyethylenephase with 1 mm, 2 mm thick are adopted during the chewing116 cycle in this paper.116

114 114 12874 76 78 80 82 –54 128–52 –50 –48 –46Occlusal –44 phase X /mm Y /mm 126 126 Teeth FigureFigure 13. 13. ChewingChewing trajectory trajectory of ofthe the chewing chewing robot robot.. 124 124 Closing phase 122 122 mm mm / / Z Z Z Z 120 120 Openning 118 118 phase

116 116

114 114 74 76 78 80 82 –54 –52 –50 –48 –46 –44 X /mm Y /mm Figure 13. Chewing trajectory of the chewing robot.

Machines 2021, 9, 171 16 of 22

Machines 2021, 9, x FOR PEER REVIEW Simulated food is adopted to conduct chewing test on the chewing robot in order to 16 of 21 obtain the occlusal force and TMJ force. Two kinds of materials silicone (SI) and polyethy- Machines 2021, 9, x FOR PEER REVIEWlene (PE) are chosen as the simulated food, as shown in Figure 14. As different thicknesses 16 of 21 of simulated food can produce different value of force, silicone with 4 mm, 8 mm and 12 mm thick and polyethylene with 1 mm, 2 mm thick are adopted during the chewing cycle in this paper.

(a) (b) Figure 14. Different simulated food chewed by the chewing robot : (a) Silicone; (b) Polyethylene. (a) (b) FigureFigureIn 14. the 14.Different experiment,Different simulated foodsimulated the chewed forces by thefood chewing in onechewed robot: chewing (a) Silicone;by the cycle (b chewing) Polyethylene. was recordedrobot: (a) onSilicone; the chewing (b) Polyethylene robot . prototype.In the experiment, At the theopening forces in one phase, chewing the cycle simulated was recorded on food the chewing was robotput between the upper jaw and theprototype. lower At jaw the opening (incisor phase, position). the simulated When food was theput between mouth the upper closed jaw to the occlusal phase, the and theIn lower the jaw experiment, (incisor position). the When forces the mouth in closedone chewing to the occlusal cycle phase, thewas recorded on the chewing robot simulatedprototype.simulated food food wasAt chewedwasthe openingchewed by the robot. byphase, Figure the 15 robot.the shows simulated a Figure silicone simulated 1 5food shows food was is a putsilicone between simulated the upper food isjaw and beingbeing chewedchewed by the by chewing the chewing robot. Different robot. simulated Different food was testedsimulated one by one. food was tested one by one. the lower jaw (incisor position). When the mouth closed to the occlusal phase, the simulatedMove forward foodand downward was chewedMove laterally by the Simulatedrobot. Figurefood Silicone 15 shows a silicone simulated food is being chewed by the chewing robot. Different simulated food was tested one by one. Move forward and downward Move laterally Simulated food Silicone

Mouth-opening status Chewing simulated food Figure 15. Picture of chewing simulated food.

Figure5.2. Experimental 15. Picture Results of of chewing Occlusal Force simulated and TMJ Force food. Once theMouth chewing-opening robot started status chewing the simulatedChewing food, both simulated the upper food jaw 5.2.structure Experimental and the condyle Results rod were of stressed. Occlusal The occlusal Force force and and TMJ TMJ forceForce expressed by strains were collected by the strain data acquisition system during the chewing course. The FigurestrainsOnce of 15. the fourthePicture sensors chewing of with chewing 1 mm robot thick simulated startedpolyethylene chewing infood one chewing. the cycle simulated are shown food, both the upper jaw in Figures 16 and 17. structure and the condyle rod were stressed. The occlusal force and TMJ force expressed by5.2. strains Experimental were collected Results by of the Occlusal strain data Force acquisition and TMJ Forcesystem during the chewing course. The strains of the four sensors with 1 mm thick polyethylene in one chewing cycle are Once the chewing robot started chewing the simulated food, both the upper jaw shown in Figures 16 and 17. structure and the condyle rod were stressed. The occlusal force and TMJ force expressed by strains were collected by the strain data acquisition system during the chewing course. The strains of the four sensors with 1 mm thick polyethylene in one chewing cycle are shown in Figures 16 and 17.

(a) (b) Figure 16. Strain on the upper jaw: (a) Lateral strain, (b) Vertical strain.

Machines 2021, 9, x FOR PEER REVIEW 16 of 21

(a) (b) Figure 14. Different simulated food chewed by the chewing robot: (a) Silicone; (b) Polyethylene.

In the experiment, the forces in one chewing cycle was recorded on the chewing robot prototype. At the opening phase, the simulated food was put between the upper jaw and the lower jaw (incisor position). When the mouth closed to the occlusal phase, the simulated food was chewed by the robot. Figure 15 shows a silicone simulated food is being chewed by the chewing robot. Different simulated food was tested one by one. Move forward and downward Move laterally Simulated food Silicone

Mouth-opening status Chewing simulated food

Figure 15. Picture of chewing simulated food.

5.2. Experimental Results of Occlusal Force and TMJ Force Once the chewing robot started chewing the simulated food, both the upper jaw structure and the condyle rod were stressed. The occlusal force and TMJ force expressed by strains were collected by the strain data acquisition system during the chewing course. Machines 2021, 9, 171 17 of 22 The strains of the four sensors with 1 mm thick polyethylene in one chewing cycle are shown in Figures 16 and 17.

Machines 2021, 9, x FOR PEER REVIEW 17 of 21 (a) (b) Figure 16. Strain on theFigure upper 16. Strain jaw: on (a the) Lateral upper jaw: strain, (a) Lateral (b) Vertical strain, (b )strain Vertical. strain.

(a) (b) Figure 17. StrainFigure on 17. theStrain condyles on the condylesof TMJs: of ( TMJs:a) Strain (a) Strain of right of right TMJ, TMJ, (b ()b Strain) Strain of of LeftLeft TMJ. TMJ.

The force can be calculated with the following methods. The material of the TMJs and The force can beupper calculated jaw is aluminum with the alloy following and the modulusmethods. of elasticityThe material of them of is the 0.7 × TMJs105 MPa. and Based upper jaw is aluminumon the alloy deﬁnitions and the of stressmodulus and strain of elasticity and Hooke’s of them Law, theis 0.7 stress × 10 can5 MPa. be obtained: Based on the definitions of stress and strain and Hooke’s Law, the stress can be obtained: σ = E · ε = 0.7 × 1011 × 12.5 × 10−6 = 8.75 × 105 (31) E    0.7  1011  12.5  10 6  8.75  10 5 (31) where σ is the stress of the upper jaw. E is the Young’s modulus of aluminum alloy and ε is where σ is the stressthe of strainthe upper shown jaw. in Figure E is 16theb. TheYoung occlusal’s modulus force can of be aluminum calculated as alloy follows: and ε is the strain shown in Figure 16b. The occlusal force can be calculated− as follows: F = σ · A = 8.75 × 105 × 1.2 × 10 4 = 105 (32) 5 4 where F isFA the occlusal  8.75 force  and 10 A 1.2is the  10 cross-sectional  105 area of the middle position(32) of where F is the occlusalstrain force gauge and on upper A is jawthe (Figure cross -12sectionala). Similarly, area the of cross-sectional the middle area position of the condylarof block is 49 × 10−6 m2 and the TMJ force can also be calculated. Multi-group experiments strain gauge on upperwere jaw carried (Figure out and 12a). the Similarly, occlusal forces the and cross TMJ-sectional forces were area also of calculated. the condylar block is 49 × 10−6 m2 andThe the results TMJ areforce shown can in also Figures be calculated.18 and 19. As Multi can be-group seen from exper theiments bar chart, the were carried out andocclusal the occlusal force increases forces with and thicknesses TMJ forces of twowere kinds also of calculated. simulated food increases directly. The results areThe shown increase in ofFigure the occlusals 18 and force 19 causes. As can the TMJ be seen forces from become the larger. bar Thechart, results the verify that the loading of the TMJ increases along with occlusal force. occlusal force increases with thicknesses of two kinds of simulated food increases directly. The increase of the occlusal force causes the TMJ forces become larger. The results verify that the loading of the TMJ increases along with occlusal force. The maximum occlusal force occurred when the robot chews the 2-mm thick polyethylene, which is 303.2 N and the corresponding left and right TMJ force is 70.0 N and 72.7 N, respectively. The minimum occlusal force is 104.2 N and corresponding left and right TMJ force is 32.9 N and 38.4 N when the robot bites the 4-mm thick silicone. The occlusal force increases by 190.98% from silicone with 4 mm thick to polyethylene with 2 mm thick. Nevertheless, the corresponding left TMJ force only increases by 120.97% and the right TMJ force only increases by 82.29%. It can be seen from the changing process of the two forces that the growth rate of occlusal force is larger than that of TMJ force.

MachinesMachines 2021 2021, ,9 9, ,x x FOR FOR PEER PEER REVIEW REVIEW 1818 ofof 2121 Machines 2021, 9, 171 18 of 22

4040 36.136.1 3535 32.632.6 3030 25 22.5 19.9 25 21.221.2 22.5 19.9 20.420.4 17.5 19.519.5 2020 17.017.0 17.5 12.5 15 12.5 12.4 15 10.3 12.4 Strain (με) 10.3 11.2 Strain (με) 10.010.0 9.69.6 11.2 1010 55 00 PEPE 1mm1mm PEPE 2mm2mm SISI 4mm4mm SISI 8mm8mm SISI 12mm12mm Occlusal force Right HKP force Left HKP force Occlusal force Right HKP force Left HKP force FigureFigure 18. 18. StrainStrainFigure ofof 18. jawjawStrain andand of HKPsHKPs jaw and ofof HKPs differentdifferent of different simulatedsimulated simulated foodfood food withwith with differentdifferent different thicknesses.thicknessesthicknesses. .

350350 303.2303.2 300300 273.8273.8 250250 189.0 200200 189.0

150150

Force(N) 105.0 104.2

Force(N) 105.0 104.2 100 100 72.7 60.0 66.9 68.3 72.7 70.070.0 58.358.3 60.0 66.9 68.3 35.3 34.3 38.438.4 5050 35.3 34.3 32.932.9

00 PEPE 1mm1mm PEPE 2mm2mm SISI 4mm4mm SISI 8mm8mm SISI 12mm12mm Occlusal force Right HKP force Occlusal force Right HKP force Figure 19. FigureFigure 19. 19. OcclusalOcclusalOcclusal forceforce force andand and TMJ TMJ TMJ (HKP)(HKP) (HKP) forceforce ofofof different differentdifferent simulated simulatedsimulated food food withfood different withwith differentdifferent thicknesses. thicknesses. thicknesses. The maximum occlusal force occurred when the robot chews the 2-mm thick polyethylene, which is 303.2 N and the corresponding left and right TMJ force is 70.0 N and 72.7 N, TheThe maximum maximumrespectively. bite bite force force The that that minimum human human occlusal can can apply apply force is to to 104.2 the the N molars molars and corresponding which which has has left been been and right measuredmeasured about about 700 700TMJ N N force o orr is higher higher 32.9 N and in in healthy 38.4 healthy N when people people the robot [49, [49, bites50],50], the approximately approximately 4-mm thick silicone. equal equal The to occlusal to human’s own weight.force While increases the by size 190.98% of the from condyle silicone withis only 4 mm about thick to20 polyethylene × 10 × 10 mm with 2[33], mm thick. human’s own weight.Nevertheless, While the the size corresponding of the condyle left TMJ is forceonly only about increases 20 × 10 by 120.97%× 10 mm and [33], the right whichwhich isis tootoo smallsmall totoTMJ burdenburden force only suchsuch increases aa largelarge by loadingloading 82.29%. comparedcompared It can be seen withwith from thethe the kneeknee changing jointjoint process ofof humanhuman of the two body.body. From From the the perspective perspectiveforces that the of of growth theory theory rate of of of mechanism, occlusalmechanism, force is redundant redundant larger than that actuation actuation of TMJ force. allows allows to to partiallypartially controlcontrol thethe internalinternalThe maximum forcesforces biteandand force thethe that actuatingactuating human can forceforce apply andand to the powepowe molarsrr consumption whichconsumption has been measured ofof actuatorsactuators cancan bebe redistributedredistributedabout 700 N byby or optimizationhigheroptimization in healthy methods.methods. people [49 ,50], approximately equal to human’s own × × Redundant actuationweight. in While masticatory the size of system the condyle may is onlyhelp about us to 20 reveal10 the10 mmphysiological [33], which is too Redundant actuationsmall to burdenin masticatory such a large system loading may compared help withus to the reveal knee jointthe ofphysiological human body. From phenomenon.phenomenon. In In this thisthe experiment, experiment, perspective ofitit shows theoryshows ofthe the mechanism, increment increment redundant of of occlusal occlusal actuation force force allowshas has priority priority to partially over over control thethe TMJTMJ foforce.rce. ItIt meansmeans thethe twotwo pointpoint--contactcontact HKPsHKPs inin thethe redundantredundant actuatedactuated systemsystem havehave aa positivepositive effecteffect onon generatinggenerating processprocess ofof chewingchewing forceforce causedcaused byby actuators.actuators. ThisThis suggestssuggests thethe forcesforces ofof thesethese twotwo TMJsTMJs playplay aa rolerole inin optimizingoptimizing thethe internalinternal forceforce ofof thethe reduredundantndant actuatedactuated system,system, enablingenabling thethe musclemuscle groupgroup toto outputoutput moremore efficientefficient occlusalocclusal forceforce duringduring chewingchewing process.process. Besides,Besides, thethe averageaverage valuevalue ofof occlusalocclusal forceforce onon incisorincisor ofof aa healthyhealthy youngyoung adultadult isis aboutabout 146.17146.17 NN [51][51].. AsAs wewe cancan seesee thethe forceforce thatthat thethe chewingchewing robotrobot cancan produceproduce inin thisthis experimentexperiment is is much much larger larger than than the the average average force. force. The The robot robot is is able able to to outputoutput enough enough occlusalocclusal force force which which can can be be potentially potentially used used in in new new dental dental materials materials test test or or food food prope propertiesrties

Machines 2021, 9, 171 19 of 22

the internal forces and the actuating force and power consumption of actuators can be redistributed by optimization methods. Redundant actuation in masticatory system may help us to reveal the physiological phenomenon. In this experiment, it shows the increment of occlusal force has priority over the TMJ force. It means the two point-contact HKPs in the redundant actuated system have a positive effect on generating process of chewing force caused by actuators. This suggests the forces of these two TMJs play a role in optimizing the internal force of the redundant actuated system, enabling the muscle group to output more efﬁcient occlusal force during chewing process. Besides, the average value of occlusal force on incisor of a healthy young adult is about 146.17 N [51]. As we can see the force that the chewing robot can produce in this experiment is much larger than the average force. The robot is able to output enough occlusal force which can be potentially used in new dental materials test or food properties test. The internal TMJ forces during chewing provides the possibility to study the performance test of TMJ prosthesis.

6. Conclusions The physical redundantly actuated chewing robot was specified in terms of six PUS linkage and two HKPs. The introduction of HKP allows the chewing robot to be able to reproduce the TMJ movement and influence of the TMJ force on occlusal force. Accordingly, it can improve the bio-imitability of the chewing robot. Unlike the traditional ways of achieving redundant actuation, this RAPR was realized through the inclusion of additional constraints to reduce DOFs, resulting in the number of actuators is larger than the number of DOFs. In order to investigate the advantage of musculoskeletal actuation redundancy in human masticatory system, such as optimal stiffness, energy efficiency and chewing forces needed for the food breakdown, this paper studies the stiffness and optimization of driving force and presents the chewing experiment of the RAPR. In stiffness analysis, the three end- effector points method was proposed for use in differential kinematics of the RAPR with mixed DOFs, mapping the reduced number of the end-effector’s generalized variables to the joint variables. Based on the dimensionally homogeneous Jacobian matrix, the stiffness simulation results confirmed the 6PUS-2HKP RAPR with two point-contact HKPs has better stiffness than the non-redundantly actuated counterpart 6-PUS PR. In optimization of driving force, Lagrange’s equations were used to derive the dynamics of the RAPR. The driving force of the RAPR was optimized by finding the minimum Euclidean norm solution. By comparison, one can get the same result that the 6PUS-2HKP RAPR with two HKPs performs better than the 6-PUS PR. These simulation results of chewing robot show the musculoskeletal structure of human masticatory system itself has the advantage of good stiffness and energy saving. They also reveal the significance of TMJs in the redundantly actuated human masticatory system. As the TMJs are modeled as two point-contact HKPs in the proposed robot, the occlusal force generated has more realistic implication. The experiments of chewing simulated food with different thicknesses are carried out to evaluate the occlusal force and the TMJ force. The occlusal force increases with the thicknesses of simulated food, and the corresponding TMJ force also increases, but at a slower rate. From a biomechanical point of view, the experiment also illustrates that the existence of TMJ in human redundantly actuated masticatory system ensures more efficient transmission and transformation from muscle force to occlusal force. Ongoing research on the robot prototype involves a wear-resistant test of the alloy ceramic crown, which evaluates the wear condition of the occlusal surfaces of the crowns. Future work may focus on developing a new generation of chewing robot. As in the current version, rigid linkages and rigid contacts are used to mimic the muscles and TMJs, respectively, the chewing force and TMJ force generated by the chewing robot are still not completely coincident with human masticatory system. Emphasis in future studies Machines 2021, 9, 171 20 of 22

will involve adopting parallel ﬂexible drive system considering the muscle elasticity, and designing soft TMJ structure considering the disc and ligament in the chewing robot.

Author Contributions: Conceptualization, H.W. and M.C.; funding acquisition, H.W. and M.C.; investigation, Y.Z.; methodology, H.W.; project administration, M.C.; software, H.W. and G.W.; supervision, M.C. and Z.Z.; validation, H.W., Z.Z. and Y.Z.; visualization, G.W.; writing—original draft, H.W.; writing—review and editing, M.C. and Z.Z. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Natural Science Foundation of China, Grant numbers 51705063 and 51575078, and the Natural Science Foundation of Jiangsu Province, China, Grant No. BK20190368. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References 1. Flores, E.; Fels, S.; Vatikiotis-Bateson, E. Chew on this: Design of a 6DOF anthropomorphic robotic jaw. In Proceedings of the 16th IEEE International Symposium on Robot and Human Interactive Communication, Jeju, Korea, 26–29 August 2007; pp. 648–653. 2. Alemzadeh, K.; Jones, S.B.; Davies, M.; West, N. Development of a Chewing Robot with Built-in Humanoid Jaws to Simulate Mastication to Quantify Robotic Agents Release from Chewing Gums Compared to Human Participants. IEEE. Trans. Biomed. Eng. 2020, 68, 492–504. [CrossRef] 3. Takanobu, H.; Maruyama, T.; Takanishi, A.; Ohtsuki, K. Mouth opening and closing training with 6-DOF parallel robot. In Proceedings of the 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation, San Francisco, CA, USA, 24–28 April 2000; pp. 1384–1389. 4. Celebi, N.; Rohner, E.C.; Gateno, J.; Noble, P.C.; Ismaily, S.K.; Xia, J.J. Development of a Mandibular Motion Simulator for Total Joint Replacement. J. Oral Maxil. Surg. 2011, 69, 66–79. [CrossRef][PubMed] 5. Wen, H.Y.; Cong MWang, G.F.; Qin, W.L.; Xu, W.L.; Zhang, Z.S. Dynamics and Optimized Torque Distribution Based Force/position Hybrid Control of a 4-DOF Redundantly Actuated Parallel Robot with Two Point-contact Constraints. Intern. J. Control Automat. Syst. 2019, 17, 1293–1303. [CrossRef] 6. Naser, M.; Jaspreet, D.; Alexander, V.; John, B.; Xu, W.L. Optimizing the torque distribution of a redundantly actuated parallel robot to study the temporomandibular reaction forces during food chewing. J. Mech. Robot 2020, 12, 1–28. 7. Kandasamy, S.; Greene, C.; Rinchuse, D.; Stockstill, J. TMD and Orthodontics; Springer: Cham, Switzerland, 2015. 8. Kishi, T.; Hashimoto, K.; Takanishi, A. Human Like Face and Head Mechanism. In Humanoid Robotics: A Reference; Goswami, A., Vadakkepat, P., Eds.; Springer: Dordrecht, The Netherlands, 2017. 9. Cid, F.; Moreno, J.; Bustos, P.; Núñez, P. Muecas: A multi-sensor robotic head for affective human robot interaction and imitation. Sensors 2014, 14, 7711–7737. [CrossRef] 10. Ding, H.; Esfahani, M.I.M.; Shen, Y.; Tan, H.; Zhang, C. Hybrid Humanoid Robotic Head Mechanism: Design, Modeling, and Experiments with Object Tracking. In Proceedings of the 2020 8th IEEE RAS/EMBS International Conference for Biomedical Robotics and Biomechatronics (BioRob), New York, NY, USA, 29 November–1 December 2020. 11. Takanobu, H.; Takanishi, A.; Ozawa, D.; Ohtsuki, K. Integrated dental robot system for mouth opening and closing training. In Proceedings of the 2002 IEEE International Conference on Robotics and Automation, Washington, DC, USA, 11–15 May 2002; pp. 1428–1433. 12. Xu, W.L.; Lewis, D.; Bronlund, J.E.; Morgenstern, M.P. Mechanism, design and motion control of a linkage chewing device for food evaluation. Mech. Mach. Theory 2008, 43, 376–389. [CrossRef] 13. Xu, W.L.; Torrance, J.D.; Chen, B.Q.; Potgieter, J.; Bronlund, J.E. Kinematics and experiments of a life-sized masticatory robot for characterizing food texture. IEEE Trans. Ind. Electron. 2008, 55, 2121–2132. [CrossRef] 14. Xu, W.L.; Pap, J.S.; Bronlund, J.E. Design of a biologically inspired parallel robot for foods chewing. IEEE Trans. Ind. Electron. 2008, 55, 832–841. [CrossRef] 15. Raabe, D.; Harrison, A.; Ireland, A.; Alemzadeh, K.; Sandy, J.; Dogramadzi, S.; Melhuish, C.; Burgess, S. Improved single- and multi-contact life-time testing of dental restorative materials using key characteristics of the human masticatory system and a force/position-controlled robotic dental wear simulator. Bioinspir. Biomim. 2012, 7, 016002. [CrossRef] 16. Papo, M.J.; Catledge, S.A.; Vohra, Y.K. Mechanical wear behavior of nanocrystalline and multilayer diamond coatings on temporomandibular joint implants. J. Mater. Sci. Mater. Med. 2004, 15, 773–777. [CrossRef] 17. Mueller, A. Redundant Actuation of Parallel Manipulators. Parallel Manipulators, towards New Applications; I-Tech Education and Publishing: Vienna, Austria, 2008. Machines 2021, 9, 171 21 of 22

18. Shang, W.; Cong, S. Dexterity and Adaptive Control of Planar Parallel Manipulators with and without Redundant Actuation. J. Comput. Nonlinear Dyn. 2015, 10, 011002. [CrossRef] 19. Ebrahimi, I.; Carretero, J.A.; Boudreau, R. 3-PRRR redundant planar parallel manipulator: Inverse displacement, workspace and singularity analyses. Mech. Mach. Theory 2007, 42, 1007–1016. [CrossRef] 20. Lee, G.; Park, S.; Lee, D.; Park, F.C.; Jeong, J.I. Minimizing Energy Consumption of Parallel Mechanisms via Redundant Actuation. IEEE/ASME Trans. Mechatron. 2015, 20, 2805–2812. [CrossRef] 21. Hufnagel, T.; Mueller, A. A projection method for the elimination of contradicting decentralized control forces in redundantly actuated PKM. IEEE Trans. Robot. 2012, 28, 723–728. [CrossRef] 22. Shang, W.; Cong, S.; Ge, Y. Coordination motion control in the task space for parallel manipulators with actuation redundancy. IEEE Trans. Autom. Sci. Eng. 2013, 10, 665–673. [CrossRef] 23. Liu, G.F.; Cheng, H.; Xiong, Z.H.; Wu, X.Z. Distribution of singularity and optimal control of redundant parallel manipulators. In Proceedings of the 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems, Maui, HI, USA, 29 October–3 November 2001; pp. 177–182. 24. Zhu, Z.; Dou, R. Optimum design of 2-DOF parallel manipulators with actuation redundancy. Mechatronics 2009, 19, 761–766. [CrossRef] 25. Marquet, F.; Krut, S.; Company, O.; Pierrot, F. ARCHI: A redundant mechanism for machining with unlimited rotation capacities. In Proceedings of the 2001 IEEE International Conference on Robotics and Automation, Seoul, Korea, 21–26 May 2001; pp. 683–689. 26. Wu, J.; Wang, J.; Wang, L.; Li, T.; You, Z. Study on the stiffness of a 5-DOF hybrid machine tool with actuation redundancy. Mech. Mach. Theory 2009, 44, 289–305. [CrossRef] 27. Wang, L.; Wu, J.; Wang, J.; You, Z. An experimental study of a redundantly actuated parallel manipulator for a 5-DOF hybrid machine tool. IEEE ASME Trans. Mechatron. 2009, 14, 72–81. [CrossRef] 28. Kim, J.; Park, F.C.; Sun, J.R.; Kim, J. Design and analysis of a redundantly actuated parallel mechanism for rapid machining. IEEE Trans. Robot. Autom. 2001, 17, 423–434. [CrossRef] 29. Zhao, Y.; Gao, F. Dynamic performance comparison of the 8PSS redundant parallel manipulator and its non-redundant counterpart-the 6PSS parallel manipulator. Mech. Mach. Theory 2009, 44, 991–1008. [CrossRef] 30. Xu, M.; Rong, C.; He, L. Design and Modeling of a Bio-Inspired Flexible Joint Actuator. Actuators 2021, 10, 95. [CrossRef] 31. Wang, W.; Hou, Z.G.; Cheng, L.; Tong, L.; Peng, L.; Peng, L.; Tan, M. Toward patients’ motion intention recognition: Dynamics modeling and identiﬁcation of iLeg—An LLRR under motion constraints. IEEE Trans. Syst. Man Cybern. Syst. 2017, 46, 980–992. [CrossRef] 32. Kim, J.; Yang, J.; Yang, S.T.; Oh, Y.; Lee, G. Energy-Efﬁcient Hip Joint Offsets in Humanoid Robot via Taguchi Method and Bio-inspired Analysis. Appl. Sci. 2020, 10, 7287. [CrossRef] 33. Takanobu, H.; Takanishi, A.; Kato, I. Design of a mastication robot using a human skull model. In Proceedings of the 1993 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS’93), Yokohama, Japan, 26–30 July 1993; pp. 203–208. 34. Vilimek, M.; Goldmann, T. Computation of forces acting in temporomandibular joint and masticatory muscles during mastication. Comput. Method. Biomech. Biomed. Eng. 2011, 14, 267–268. [CrossRef] 35. Tanaka, E.; Koolstra, J.H. Biomechanics of the temporomandibular joint. J. Dent. Res. 2008, 87, 989–992. [CrossRef][PubMed] 36. Mercuri, L.G. Temporomandibular Joint Total Joint Replacement- TMJ TJR; Springer International Publishing: Cham, Switzerland, 2016. 37. Iwasaki, L.R.; Baird, B.W.; McCall, W.D., Jr.; Nickel, J.C. Muscle and temporomandibular joint forces associated with chincup loading predicted by numerical modeling. Am. J. Orthod. Dentofacial Orthop. 2003, 124, 530–540. [CrossRef] 38. Nickel, J.C.; Iwasaki, L.R.; Walker, R.D.; Mclachlan, K.R.; McCall, W.D., Jr. Human masticatory muscle forces during static biting. J. Dent. Res. 2003, 82, 212–217. [CrossRef] 39. Koolstra, J.H. Dynamics of the human masticatory system. Crit. Rev. Oral Biol. Med. 2002, 13, 366–376. [CrossRef][PubMed] 40. Koolstra, J.H.; Eijden, T.M.G.J.V. Prediction of volumetric strain in the human temporomandibular joint cartilage during jaw movement. J. Anat. 2006, 209, 369–380. [CrossRef][PubMed] 41. Park, J.T.; Lee, J.G.; Won, S.Y.; Lee, S.H.; Cha, J.Y.; Kim, H.J. Realization of masticatory movement by 3-dimensional simulation of the temporomandibular joint and the masticatory muscles. J. Craniofacial Surg. 2013, 24, 347–351. [CrossRef] 42. Iwasaki, L.R.; Petsche, P.E.; McCall, W.D., Jr.; Marx, D.; Nickel, J.C. Neuromuscular objectives of the human masticatory apparatus during static biting. Arch. Oral Biol. 2003, 48, 767–777. [CrossRef] 43. May, B.; Saha, S.; Saltzman, M. A three-dimensional mathematical model of temporomandibular joint loading. Clin. Biomech. 2001, 16, 489–495. [CrossRef] 44. Koolstra, J.H.; Eijden, T.M.G.J.V. Dynamics of the human masticatory muscles during a jaw open-close movement. J. Biomech. 1997, 30, 883–889. [CrossRef] 45. Koolstra, J.H.; Eijden, T.M.G.J.V. A method to predict muscle control in the kinematically and mechanically indeterminate human masticatory system. J. Biomech. 2001, 34, 1179–1188. [CrossRef] 46. Wen, H.; Xu, P.; Cong, M. Kinematic Model and Analysis of an Actuation Redundant Parallel Robot With Higher Kinematic Pairs for Jaw Movement. IEEE Trans. Ind. Electron. 2014, 62, 1590–1598. [CrossRef] 47. Merlet, J.P. Jacobian, manipulability, condition number and accuracy of parallel robots. J. Mech. Des. 2005, 128, 199–206. [CrossRef] Machines 2021, 9, 171 22 of 22

48. Pulloquinga, J.L.; Escarabajal, R.J.; Ferrándiz, J.; Vallés, M.; Mata, V.; Urízar, M. Vision-Based Hybrid Controller to Release a 4-DOF Parallel Robot from a Type II Singularity. Sensors 2021, 21, 4080. [CrossRef] 49. Ferrario, V.F.; Sforza, C.; Zanotti, G.; Tartaglia, G.M. Maximal biteforces in healthy young adults are predicted by surface electromyography. J. Dent. 2004, 32, 451–457. [CrossRef] 50. Okiyama, S.; Ikebe, K.; Nokubi, T. Association between masticatory performance and maximal occlusal force in young men. J. Oral Rehabil. 2003, 30, 278–282. [CrossRef] 51. Ferrario, V.F.; Sforza, C.; Serrao, G.; Dellavia, C.; Tartaglia, G.M. Single tooth bite forces in healthy young adults. J. Oral Rehabil. 2004, 31, 18–22. [CrossRef]