Dynamic modelling and control of an underactuated Klann-based

Edgar A. Mart´ınez-Garc´ıa* and Josue´ Dom´ınguez Roman Lavrenov Laboratorio de Robotica,´ Institute Engineering and Technology Laboratory of Intelligent Systems, ITIS Universidad Autonoma´ de Ciudad Juarez´ Kazan Federal University Juarez,´ Mexico Kazan, Russia [email protected] [email protected]

Abstract—This work discloses the dynamic control model of consumption demand rises as the number of actuators in- a hexapod robot with tripod-based walking gait. The proposed creases [22], which drastically reduces the robot’s operating walking model is based on the Klann and three main time. Traditionally, multi-legged robots may require at least actuators to provide quasi-omnidirectional mobility. The reduced number of actuators preserve holonomy as similar as popular two actuators (2 DOF) to control one leg, one for azimuth 18-servo hyper-redundant hexapods (three servos per leg). This rotation and another for elevation. In addition, incrementing work proposes two-drive differential control, one drive per the number of actuators to control a limb, the control models lateral triplet of legs. Each triplet is synchronized in speed with become redundantly kinematic. The number of independent different angle phase of rotation. The six limbs are synchronized control variables is greater in number than the DOF in the with bidirectional yaw motion with the third actuator. Quasi- omnidirectional mobility was achieved and controlled by a working space. Usually walker robots control is devoted for dynamic control law that governs the robots mechanisms motion. cyclod gaiting control [13], resembling biomimetic gaits [18], Kinematic and dynamic results are validated through numerical gaits synthesis [5], synthesis for limbs reconfiguration [4], gait simulations using a tripod gait. planning [10] and learning [11]. Index Terms—hexapod, multi-legged, Klann limb, dynamic control, underactuated mechanism, walker robot, tripod gait

I.INTRODUCTION Multi-legged walker robots inherently pose great locomo- tion capabilities due to their hype-stability over numerous complex reliefs. Some unstructured terrains include flatter, alluvial plains, farming soils, steeper, rockier uplands, wa- tershed boundaries, drainage characterized systems and hilly terrains [1]. For human safety, multi-legged walker robots [2] are suitable to operate at dangerous landscapes where a) landslides, downhill creep, flows, slumps, and rock falls rep- resent hazardous environments for humans [3]. There exist different classes of robotic hexapods developed to perform different varieties of missions and tasks that are of great utility in field applications, [12]. Multi-legged robots poses high steerage, which is essential to carry out missions in fields such as mining, forestry, construction, planetary exploration, vulcanography, search and rescue, demining, agriculture and so forth. There are few successful applications based on undearctuated limbs for walking such as trajectory tracking quadrupeds [15], bio-inspired hexapods [20] [14], reconfigurable stair climbing walkers [6], walking chairs [8], or similar works related to the present research [7]. Redundant walker robots such as hexapods [16] [19] require one rotary b) actuator per degree of freedom (DOF), which is a general Fig. 1. Proposed quasi-omnidirectional robotic platform. a) Kinematic param- complication. The joints require wider physical space, the eters (top view). b) Underactuated three-motor quasi-omnidirectional robot joints mechanism require instrumentation, electrical energy concept.

978-1-7281-3021-7/19/$31.00©2019 IEEE In this work’s concept, an asymmetric hexapod deploying Klann limbs is proposed (Figure 1a). The Klann linkage is a planar mechanism with 7 passive joints, using one rotary actuator, but in this work the mechanism is also be controlled in yaw. One of our work’s main contributions is the mechanical design that reduced a traditional hyoer-redundantn eighteen- servo model into an asymmetric three-motor underactuated version of hexapod, still preserving omnidirectional mobility (Figure 11b). One servomotor for limbs bidirectional syn- chronized steering, a second motor for driving the right-sided triplet of legs, and a third motor to drive the left-sided triplet. As a difference from other approaches [17], the proposed hexapod design is considered quasi-omnidirectional because a) it can reach any Cartesian point requiring a small latency time for the limbs to reach the targeted yaw. A dynamic control law is presented for the gait speed by establishing a reference torque to track a given trajectory. Each lateral triplet has the limbs position with different phase to stabilize the gait. The limbs dynamic motion model includes the kinetic and potential energy interaction model, which approaches a Euler-Lagrange solution. The presented approach provides simulations that concern the underactuated mobility space of a hexapod. In section II, the proposed underactuated mechanical design and the Klann limb kinematics are presented. In section III presentes the deduction of the dynamic control walking b) model and section IV provides the conclusion. Fig. 2. Differential drive and steering mechanisms. a) driving structure. b) II.KLANN-BASED UNDERACTUATED KINEMATICS Lims yawing structure. The present work proposes a locomotion mechanism con- sisting of a differential driving system (Figure 2a), an all-limb synchronous bidirectional steering yaw mechanism (Figure 2b) the optimize the link-foot linearity of motion with half rotation and the Klann-based limbs (Figure 3). The right and left sided of the crank handle (Figure 3b). The rest of the crank’s rotation allows that the link-foot move up to a defined height are differential speeds providing instantaneous velocity vt and before motion return toward the staring position, repeating yileding instantaneous yaw speed, ωt. One drive per lateral triplet of limbs, interconnected front/back sides by tracks from again the motion loop. In this work, a Klann mechanism of the central drives. The steering mechanical system of Figure 2b shows bilateral direction angle for all-limb synchronously and can turn in yaw −π/4, ..., π/4. The Klann mechanism (Figure 3) is an underactuated planar multi-link system, which from rotary input motion, it produces as output a cycloid trajectory (Figure 4). The links proportions are defined to optimize linear motion of the contact point at every rotary half cycle of the crank. The contact point lifts during the other rotary half cycle, before returning to the staring position. Klann [9] presented his famous invention, the ”walking device”, which is usually deployed to emulate biological limbs motion, a) b) mostly antropods. For instance, [21] developed artificial active whiskers for underwater guidance of walking robots based on Fig. 3. Klann-based limb. a) free-body diagram. b) Klann limb with motor. the Klann mechanism. The Klann mechanism has several func- tional advantages as numerous advanced displacement systems specific proportions was designed for the purpose of limbs in have, such as stepping over obstacles, climb stairs, walking a tripod-like walking. The kinematic model was deduced to over all terrains, while not requiring a computing system to numerically simulate and analyze the gait tracks. The link L1 be controlled. The Klann mechanism is comprised of seven in connection with L3 actuate as a rotating cam with L2 as rigid links (L1, ..., L7, Figure 3a), a stretcher frame, a link a balancing bar. The oscillatory motion is transmitted to the used as crank handle L1, two as seesaw that are joined to the coupled links L4, L5 and L6 to convert the rotary motion of stretcher frame (L2 and L5), and all of them are interconnected L1 into linear motion in the base of the link effector L7. Let through pivot joints (A, B, ..., F ). The links proportions define us define the crank handle Cartesian position model of link L1: It follows that, by using the segments L5, L6 and R2 within Ax = L1 cos(φ0) (1a) the law of Cosines, the inner angle β2 is obtined by  2 2 2  and −1 R2 + L6 − L5 β2 = cos . (12) Ay = L1 sin(φ0), (1b) 2R2L6 where φ0 is the motor’s rotation angle conducted to the link The angle φ4 determines ths L6 slope w.r.t. node F , L1. The instantaneous distance R1 between the passive joints A and B is calculated by φ4 = (α2 − β2) − π. (13) q By using the links L , L , L y L the instantaneous position R = (A − B )2 + (A − B )2, (2) 1 3 4 6 1 x x y y of node E is obtained, forming a non stationary angle α1 of the straight line AB with Ex = L1 cos φ0 + L3 cos φ2 + L4 cos φ3 + L6 cos φ4 (14a) respect to (w.r.t.) the horizontal axis, such that   and −1 Ay − By α1 = tan , (3) Ax − Bx Ey = L1 sin φ0 + L3 sin φ2 + L4 sin φ3 + L6 sin φ4, (14b) with complementary angle β1 that comprises the triangle ABC with the coordinates of joints D and E known, the angle φ5 and is obtained through the law of Cosines: is calculated with the expression:  2 2 2  −1 R1 + L2 − L3  E − D  β1 = cos . (4) φ = tan−1 y y . 2R1L2 5 (15) Ex − Dx Therefore, the angle φ governs the motion of L and is 1 2 In addition, to determine the gait angular profile, the angle inferred by slope angle ∆φ2 is included, φ1 = (α1 − β1) − π. (5) φ6 = φ4 + ∆φ2, (16) Thus, through the angle φ1, the Cartesian position of the node

C is deduced by the expressions where ∆φ2 = 90.6 · π/180 + π/2. Finally, the node G represents the Klann limb’s end effector Cx = Bx + L2 cos(φ1) (6a) and Gx = L1 cos φ0 + L3 cos φ2 + L4 cos φ3 + L7 cos φ6 (17a) Cy = By + L2 sin(φ1), (6b) and where Bx and By describe the L2 coordinates. By knowing Gy = L1 sin φ0 + L3 sin φ2 + L4 sin φ3 + L7 sin φ6. (17b) the positions of nodes A and C, the angle φ2 can be obtained using Therefore, by computing the previous kinematic formulation,   −1 Cy − Ay the joints A, . . . , F and limb’s contact point G are numerically φ2 = tan . (7) Cx − Ax simulated as shown by Figure 4.

Therefore, as for the angle φ3 its model is:

φ3 = φ2 + ∆φ1, (8) where ∆φ1 = 160.53·π/180+π being the slope angle of L3. Thus, in order to model the position of joint F , the following expression is established where the links L1, L3 and L4 are involved in describing the φ2 behaviour. Thus,

Fx = L1 cos φ0 + L3 cos φ2 + L4 cos φ3 (9a) and Fy = L1 sin φ0 + L3 sin φ2 + L4 sin φ3. (9b)

Furthermore, to find the magnitude of R2, the positions of joints D and F are used in the following expression: q 2 2 R2 = (Fx − Dx) + (Fy − Dy) . (10)

Moreover, the angle α2 is obtained w.r.t. the absolute horizon- tal plane by   Fig. 4. Numerical simulation of the Klanns joints Cartesian motion. −1 Fy − Dy α2 = tan . (11) Fx − Dx III.WALKING CONTROL LAW There is one pair of Cartesian increments ∆x and ∆z for each limb defined by translation coordinates [(∆x, ∆z), The equations presented here are used to infer the robot’s (∆x, −∆z), (0, −∆z), (0, ∆z), (−∆x, ∆z), (−∆x, −∆z)], state vector and its higher order derivatives about its locomo- w.r.t. the center of the hexapod. The Euler rotation matrix tion overtime. The independent control joint variables are the Rz transforms each limb i, and their end-effectors position orientation angles ϕ˙ s, which governs the Klann rotary shaft vectors pi. In addition, the robots longitudinal and angular positions. The limbs’ end-effectors kinematic state vector is speed models described by the control vector is ˙u = (v, ω)>. p = (p , p , p )> sin(ϕ ) s ,..., cos(ϕ +ϕ ) c x y z , with 1 , 1 1 2 , 12 The absolute robots velocity is modeled by the averaged lateral p and so forth, in order to simplify expressions. Thus, stating limb Cartesian speeds by v + v k ˙q + ˙q k p2 (x ˙ +x ˙ )2 + (y ˙ +y ˙ )2  >          v = r l = ir il = r l r l ss 0 0 c0 c2 c3 c6 t 2 2 2 p =  0 1 0  l1 s0 + l3 s2 + l4 s3 + l7 s6 (20) 0 0 cs c0 c2 c3 c6 where vt is the robot’s instantaneous velocity, vr and vl (18) are the limbs’ instantaneous lateral speeds at right and left sides respectively. The Klann limb direct solution Let us establish each limbs inertial frame into the robots fixed model is defined by ˙p = J · Φ˙ , where J is Jacobian coordinate system located ideally at its geometric center > matrix and the joint angles Φ = [φA, φC , φE, φF , φG] are     −π ∆x the independent control variables. Thus, it is expressed: qi = Rz · pi + (19) 2 ∆z

  l1c0cs + l3c2cs + l4c3cs + l7c6cs −l1s6ss −l3s2ss −l4s3ss −l7s6ss ˙ ˙p =  0 l1c0 l3c2 l4c3 l7c6  · Φ. (21) −l1c0ss − l3c2ss − l4c3ss − l7c6ss −l1s0cs −l3s2cs −l4s3cs −l7s6cs

Likewise, the inverse algebraic solution is expressed by

 −1 l1c0cs + l3c2cs + l4c3cs + l7c6cs −l1s6ss −l3s2ss −l4s3ss −l7s6ss ˙ Φ =  0 l1c0 l3c2 l4c3 l7c6  · ˙p, (22) −l1c0ss − l3c2ss − l4c3ss − l7c6ss −l1s0cs −l3s2cs −l4s3cs −l7s6cs

where the pseudoinverse matrix J+ = (J · J>)−1 · J is an Therefore, the control vector ˙u is algebraically deduced with invertible, non singular and non stationary matrix. Moreover, time-variant control matrix K, the robots lateral speeds are defined by    vr vl   1 1    v 2 + 2 2 2 vr v5 + v1 k ˙q5 + ˙q1k = 2b = 2b 2b · vr = = , o bien vl = k ˙q4k (23) ω a2+b2 (vr − vl) a2+b2 − a2+b2 vl 2 2 t t t t t t | {z } K and (27) v + v k ˙q + ˙q k v = 2 6 = 2 6 , or v = k ˙q k (24) In addition, the functional forms of vr and vl are established l l 3 > 2 2 by the general expression λq(φ0, φs) , which is in terms of From Figure 1a, the robot’s angular velocity is described by the robots yaw orientation, such that 2b(v − v ) ω = r l (25)  k ˙q (φ˙ , φ˙ )k  2 2   1 0 s at + bt v r  .  = λt ·  .  (28) Considering that at and bt are non constant as legs yaw vl k ˙q (φ˙ , φ˙ )k φs changes, where at = ||q5 − q1|| and bt = |zr − zl|. 6 0 s Moreover, z = E cos(φ ) and z = E cos(φ ), where E is the r s l s Where λ is the commutation gait that switches either the right instantaneous length from the Klanns rotary point to the end- or the left sided tripod, and exchanging the value of the tripod’s effectors contact point. When the angle φ = 0, then b keeps s t potential energy P or P supporting the robots weight w/3 constant and such an angle is aligned to robots longitudinal r l at each of the three contact points, defined by axis, and producing as a result the following specific case: ( (k ˙q5k+k ˙q1k)−(k ˙q2+k ˙q6k) P 2|zr − zl|( ) λ1,Pr = hg mi + w/3 ω = 2 (26) λ = i 2 2 P (kq5k − kq1k) + |zr − zl| λ2,Pl = hg i mi + w/3 The tripod gait commutation is triggered between matrices λ1 and λ2 at every time a leg of either tripod completes a cycle, inferred by the coordinate (Gx, Gy) of expression (17). The approach in this work is constrained by two matrices provided as a general model solution that even may be used for up to six actuators (one drive per leg). Each row represents the right- side (row 1) and the left-side (row 2) of the robot’s limbs. Each λ1,2 records the desired user gait, for any matrix element λ1,2(I, j) 6= 0 establishes a contact point currently stepping ground. If a contact point does physically not match the λ1,2 in turn, then the drive speed up/down to match it. Hence, the following matrices describe both commuting walking tripods:

1/2 0 0 0 1/2 0 λ = , 1 0 0 0 1/2 0 0 (29) 0 0 1/2 0 1/2 0 λ = . 2 0 1/2 0 0 0 0 Without loss of generality, by commuting both tripods se- quentially, Figure 7ab illustrates the limbs Cartesian behaviors along the y component (Figure 5). The hexapod walking tra- Fig. 6. Robot’s mobility nearly full holonomy space, when vR > vL (dots), when vR < vL (cross), when vR = vL (circle) at different θt.

where, in the previous reference trajectory model defines the transverse trajectory included in the inverse kinematics control that has been deduced from (30), −1  ref ˆ  µt+1 = µt + K ˙u − u˙ t (31) From the reference model, the vector of lateral speeds µ˙ = > (vr, vl) is obtained. Thus, the right-sided tripod is con- strained by two legs same speeds || ˙p1|| = || ˙p5|| = vr and || ˙p4|| = vl, as well as the left-sided tripod speeds, || ˙p2|| = || ˙p6|| = vl and || ˙p3|| = vr. Thus, the contact point Cartesian velocities are v2 v2 ˙p = r , ˙p = v2 and ˙p = l , ˙p = v2. 1,5 2 4 l 2,6 2 3 r By using the inverse solution of the Klann kinematics of Fig. 5. The six legs tripod walking behavior. Vertical limbs position. previous expression (38), the speed contact points are known values, and the linkage vector of angular speeds Φ = jectory control and the walking reachable spaces with different T (φA, φC , φE, φF , φG) is inferred, and of very particular robot lateral speeds at three different all-limb orientations interest is the Klann s shaft/crank angular speed along forward and backward motions are shown in Figure −1   6. From the local Cartesian origin, any trajectory depicted Φ˙ t+1 = Φ˙ t + J ˙pt − J · Φ˙ t (32) below refers to backward navigation. Likewise, any trajectory ˙ depicted above the Cartesian origin refers to the robots forward At this point, only φA is of interest because represents the motion. Figure 6 also illustrates when the hexapods lateral angular velocity of the Klann limb’s shaft (conducted by a speeds are equal, and no angular speed is yielded, each straight kinematic chain). The three kinematic chain rotary speeds ˙ ˙ ˙ trajectory is produced at different all-limb angles at forward are φAr, φAl, φAs, right driving, left driving and steering, and backward motion. respectively. For simplicity, the three rotary variables are ˙ ˙ ˙ > Now, assuming a trajectory reference model to be tracked represented by the control vector Ω = (φr, ϕ˙ l, φs) . From by the robot, let us define such a reference trajectory vector dynamic equation (51), the following direct dynamic solution > with two components, both in terms of the first derivative as for torques τ = (τr, τl, τs) is deduced, functions of time,  P r  rr i mi 0 0  ref   2  P l ˙ ref υ a0 + a1t + a2t + ... τ =  0 rl i mi 0  · Ωt (33) ˙u = ref = 2 , (30) P s ω b0 + b1t + b2t + ... 0 0 rs i mi Thus, the general main recursive dynamic control equations: REFERENCES [1] HyunGyu, K. DongGyu, L., KyungMin, J., TaeWon S. (2016). 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