Trajectory Planning for the Five Degree of Freedom Feeding Robot Using Septic and Nonic Functions

International Journal of Mechanical Engineering and Robotics Research Vol. 9, No. 7, July 2020

Priyam A. Parikh Mechanical Engineering Department, Nimra University, Ahmedabad, India Email: [email protected]

Reena Trivedi and Jatin Dave Email: { reena.trivedi, jatin.dave}@nirmauni.ac.in

Abstract— In the present research work discusses about Instead of planning the trajectory using Cartesian scheme, trajectory planning of five degrees of freedom serial it is better to get the start point and end point of each joint manipulator using higher order polynomials. This robotic [2]. After getting the joint angles from the operational arm is used to feed semi-liquid food to the physically space, trajectory planning can be done using joint space challenged people having fixed seating arrangement. It is scheme. However trajectory planning in the Cartesian essential to plan a smooth trajectory for proper delivery of food, without wasting it. Trajectory planning can be done in space allows accounting for the presence of any constraint the joint space as well as in the cartesian space. It is difficult along the path of the end-effector, but singular to design trajectory in Cartesian scheme due to configuration and trajectory planning can’t be done with non-existence of the Jacobian matrix. In the present research operational space [3]. In the case of inverse kinematics no work, trajectory planning is done using joint space scheme. joint angle can be computed due to non-existence of the The joint space scheme offers lower and higher order inverse of the Jacobian matrix. That is the reason behind polynomial methods for the trajectory planning. In the selecting the joint space scheme for the trajectory planning present work parabolic and cubic functions are considered th th [4]. as lower polynomials and septic (7 order) and nonic (9 Trajectory planning using joint space scheme can be order) functions are considered as higher order polynomials. Lower order polynomial does not have any control over joint done in many ways. General methods are linear function, acceleration and velocity, which leads a servo actuator parabolic function, cubic functions and quintic function. towards instantaneous velocity and infinite acceleration. Above mentioned methods belong to lower order This phenomenon can cause loss of food in the delivery root, polynomial. This paper discusses about the trajectory lesser battery life, wear and tear in the joints and increases planning using septic (7th order polynomials) and nonic the probability of damaging the servo actuator. To address (9th order polynomials). Lower orders polynomials are this problem, the proposed research work presents the sometimes not desired due to its discontinuities in joint methodology and trajectory planning of a serial manipulator rates. Furthermore use of lower order polynomials in using septic and nonic functions. The higher order trajectory planning may lead robot to nonzero acceleration polynomials provide zero acceleration and velocity at the beginning and at the end. It also gives the continuity in the and velocity values at the beginning and at the end points displacement, velocity and acceleration, which is necessary of the trajectory. Sometimes use of first and second order for smoother delivery of the food.  polynomial may force servo actuator to provide instantaneous velocity and infinite acceleration, which Index Terms—Feeding robot, trajectory planning, higher leads servo motor towards locking or burning. Above order polynomials, Lower order polynomials, septic function, mentioned conditions are not acceptable for this type of nonic function, Physically challenged people. robot which carries food for patient. Shuang Fang [5] has planned a trajectory using seventh I. INTRODUCTION order polynomial method. That method was applied on a seven degree of freedom robot. He got a smooth trajectory Trajectory planning of a robot is to generate function response compare to lower order polynomials, but he had according to which a robot’s joint will move. Trajectory not shown the problem of acceleration and velocity over planning can be done using Cartesian scheme as well as shoots. Jiayan Zhang and Qingxi Meng [6] developed joint space scheme [1]. However Cartesian scheme or trajectory using improved genetic algorithm. Using this inverse kinematics is essential to perform before getting method they were able to reduce the cycle time of the robot into joint space scheme. Inverse kinematics mainly involves end-effector potion, its orientation and derivatives. as well as the trajectory got improved. Huang T. [7] applied the seventh-order B-spline curve for the trajectory planning, which achieved the optimal planning goal and the angular Manuscript received July 17, 2019; revised May 24, 2020.

displacement, velocity and acceleration curve of each joint TABLE I. HARDWARE DETAILS of the robot are smoother. However spline curves are the Details of Robot used in this paper Sr. Details combination of higher order and lower order polynomials No. Particular to achieve the replica of the exact trajectory. Xiaojie Zhao Remarks and Maoli Wang [8] planned the robot trajectory using 1 Degreeof Freedom 5 quintic polynomial method, which is nothing but a fifth 2 Type of Robot T-R-R-R-R order polynomial method. However this method could be Metal Gears, 15kg/cm torque 3 Servo Motors X 5 useful when, initial and final angular velocities and angular (Stall torque) 0.13 sec/ 60 degree at 7.2 volts acceleration are nonzero values. 4 Working Speed (No load) The novelty of this research is; the robot trajectory is 5 Working Voltage 4.8 volt-7.2 volts developed using seventh order and ninth order polynomials. In addition of angular displacement; angular velocity and 6 Controller Used ARDUINO MEGA acceleration are also given same the importance. The paper 7 Battery LI-PO 7.2 volt also discusses the merits and demerits of lower and higher Gyroscope MPU6050 for 8 Sensor used order polynomials. Acceleration The first phase of the paper shows the basic details of B. Problem Description, Methodology and Work Flow robot, hardware set up, a 3D CAD model, forward and As discussed in the introduction part, lower order inverse kinematics. However forward and inverse polynomial method does not provide any control over kinematics are not necessary to show, but before discussing acceleration and velocity. However some researchers and about the trajectories, velocity and acceleration, it becomes text book authors has proposed a linear trajectory with extremely important to discuss about forward and inverse parabolic blend, which is able to solve the problem of kinematics. In addition with that forward kinematics gives infinite acceleration. But this method contains linear information about the end-point of the robot (In this case trajectory, which makes angular acceleration almost zero end point is fixed). Based on this end-point in the Cartesian and makes angular velocity maximum in the middle space, joint angles can be found (which is called inverse portion. So above mentioned method is also not desirable kinematics). The joint angles are found from inverse as far as the case of feeding robot is concerned. Then we kinematics, can be rectified using joint space. tried to incorporate full cycloid trajectory based on cam The second phase of the paper discusses about and follower, but unfortunately it gave couple of methodology of generating trajectories using septic and acceleration and deceleration peaks. Next we replaced nonic functions. In addition with that, all the trajectories linear trajectory with exponential function. it worked well are compared and plotted along with velocity and in the case of smoothness, but somehow, it got failed to acceleration. provide zero acceleration and velocity at the end of the trajectory. At last we planned trajectory using seventh and A. Hardware Setup of the System ninth order polynomial methods. The purpose of zero According to Fig. 1 (left), the robot is vertically acceleration and velocity at the beginning and end point of downward and includes one twisting joint along with four the trajectory is solved along with smother trajectory. revolute joints. The kinematic chain diagram is shown in II. KINEMATIC MODELING USING DH MATRIX Fig. 1 (right) along with the 3D model of the robot. In the kinematic chain diagram, the base and the spoon Forward kinematics is for determining the position and (end-effector) are denoted as B and G respectively. All the orientation of a robot end-effector with respect to a hardware details are shown in Table I. reference coordinate system. In this case the joint variables and the arm parameters are already defined. Inverse kinematics of a robot manipulator deals with the calculation of each joint variable, given the position and orientation of end effector [9]. Forward kinematics is done using DH matrix frame assignment, where the methodology is explained to find end point of the end-effector using known Joint angle)twisting angle)a (Link length) and d (joint distance) A. Forward Kinematics Using DH Matrix Method In (1), F is the function of , by putting the value of , values of desired position and rotation can be found [10]. Here θ1, θ2, θ3, θ4, and θ5 are the input variables and x,y,z and R are the desired position and rotation respectively.

The robot arm parameters are shown in Table II, which are

generated using DH matrix frame assignment shown in Fig. Figure 1. Hardware Setup and 3D CAD model along with kinetics diagram 2. F(θ1, θ2, θ3, θ4, θ5) = [x, y, z, R] (1)

2 The forward kinematics assignment helps in finding the 3T = Rot( ̂Z, θ3)Trans( ̂X,L2) (6) transformation of end-effector with respect to origin, which is shown in (2). Matrix shown in (3) is a transformation 3 4T = Rot( ̂Z, θ4)Trans( ̂X,L3) (7) matrix, which is a combination of rotation matrix (3X3) and orientation (3X1) matrix. Equations 4 to 8 are derived 4 5T = Rot( ̂Z, θ5) Trans( ̂X,L4) (8) by DH matrix frame assignment along with screw principle of rotation and translation. Equation 9 is a reference matrix  CaSSCSC   to find all transformation matrices of the consecutive links. i−1 i ii iiii iT =   (9) Based on (9), all the transformation matrices are i ii SaSCCCS  iiii   calculated for each joint. Referring Table II and putting all  S0  i  i dC i    the initial values of each joint (column 1) in (9), we get the  00 10  initial transformation matrix of the robot shown in (10). Similarly putting all the final values of each joint in   001 130 0 (column 2) (9), we get the final transformation matrix, 5T =   0100    (10) shown in (11). It should be noted that (Px,Py,Pz) is the   010  320   positional vector, whereas v11 to v33 is the rotational 1000 vector.  

 .0 9848 .0 1736 0 332 8.  TABLE II. ROBOT ARM PARAMETERS (FORWARD KINEMATICS) 0 5T =  00  01       d  .0 1736 .0 9848 0 102  (11) twistin a (Link (Joint angle) (Joint angle) (joint   g length) 1000 Initial in Final in distance   mm Radian Radian )mm angle) Radian   50  0

  0 0 140 (L1)

  0 0 120 (L2)

  0  100(L3)

  0 0 80(L4)

0T =0 T1T2T03T4T (2) 5 1 2 3 4 5  Pvvv  0 1211 13 x 5T =   Figure 3. Initial and final postion of the robot in XYZ plane before and 2221 23 Pvvv y   (3) after perfomaing DH frame assignment  3231 33 Pvvv z    1000   B. Inverse Kinematics of the Robot Inverse kinematics of a robot manipulator deals with the 0 ̂ ̂ (4) 1T = Rot( Z, θ1)Rot( X, −90 ) calculation of each joint variable, given the position and 1 ̂ ̂ orientation of end effector. The computation of the inverse 2T = Rot( Z, θ2)Trans( X,L1) (5) kinematics of robot manipulator is quite difficult if compared to the forward kinematics because of the nonlinearities and multiple solutions involved [12]. Inverse kinematics helps in finding modified joint angles, founded from forward kinematics. It should be noted that, C2345 = cos (θ2 + θ3 + θ4 + θ5) and S234 = sin (θ2 + θ3 + θ4) . Here the positional vector is the function of joint variables, shown in (12). The positional vectors can be given by (14) to (16) in XYZ plane. It can be found using the product of overall transformation matrix and inverse of first transformation matrix.

F(x, y, z, R) = [θ1, θ2, θ3, θ4, θ5] (12)

0 −10 1 2 3 4 [1T] 5T =2 T3T4T5T (13)

Figure 2. DH Matrix Frame Assignemnts on the current robotic system

A. Trajectory Planning Using 7th Order Polynomial Angular displacement with respect to time with the order of seven is shown in (17). Since nth order polynomial would have n+ 1 coefficient, seventh order polynomial would have 8 coefficients. It’s first, to fourth time derivatives are given in (18), (19) and (20) respectively. Here the first derivative represents angular velocity, second time derivative represents acceleration, whereas third and fourth derivative represents jerk and jounce respectively. All the initial and final conditions at time t=0 seconds and t=tf (final time) are shown in (21) to (24) respectively. All eight unknowns are found using AX=B (shown in 25). All the initial conditions, initial angles, final angles and all coefficients are shown in Table IV. In (27), the

Figure 4. Final postion of the robot in XYZ plane after inverse displacement function with respect to time is shown for all kinematics the joints. It should be noted that jounce is not discussed in this case. Eq. 26 to 31 represents all unknown coefficients. Eq. (31) represents generalized equation for displacement PX = C1(L1C2 + L2C23 + L3C234 + L4C2345) (14) with all initial and final conditions. All the initial

Py = S1(L1C2 + L2C23 + L3C234 + L4C2345) (15) conditions for all the joints are shown in Table V. Coefficient values are shown in Table IV for septic

Pz = 50 − (L1S2 − L2S23 − L3S234 − L4S2345) (16) function for all the joints. 휃(푡) = 푎 + 푏푡 + 푐푡2 + 푑푡3 + 푒푡4 + 푓푡5 + 푔푡6 + ℎ푡7 (17)

After performing the inverse kinematics, the final 휃(푡)̇ = 푏 + 2푐푡 + 3푑푡2 + 4푒푡3 + 5푓푡4 + 6푔푡5 + 7ℎ푡6 (18) position of the robot is shown in Fig. 4.All the joint angles, calculated using IK, are shown in Table III. 휃(푡)̈ = 2푐 + 6푑푡1 + 12푒푡2 + 20푓푡3 + 30푔푡4 + 42ℎ푡5 (19)

TABLE III. ROBOT INVERSE KINEMATIC ANALYSIS 휃(⃛푡) = 6푑 + 24푒푡 + 60푓푡2 + 120푔푡3 + 210ℎ푡4 (20)

i f i f (Joint (Joint (Joint (Joint 휃(푡) Joints  angle) angle) angle) angle) 휃푖 = 푎, 푡 = 0 푠푒푐 Initial in Final in Initial in Final in = { 2 3 4 5 6 7 휃푓 = 푎 + 푏푡푓 + 푐푡푓 + 푑푡푓 + 푒푡푓 + 푓푡푓 + 푔푡푓 + ℎ푡푓 , 푡 = 푡푓 푠푒푐 Radian Radian Degree Degree (21) Joint 1     ̇ Joint 2     휃(푡) 휃̇ =푏, 푡=0 푠푒푐 = { 푖 Joint 3     ̇ 2 3 4 5 6 휃푓 = 푏 + 2푐푡푓 + 3푑푡푓 + 4푒푡푓 + 5푓푡푓 + 6푔푡푓 + 7ℎ푡푓 , 푡 = 푡푓 푠푒푐 Joint 4     (22) 휃(푡)̈ Joint 5     휃̈ = 2푐, 푡 = 0 푠푒푐 = { 푖 ̈ 2 3 4 5 휃푓 = 2푐 + 6푑푡푓 + 12푒푡푓 + 20푓푡푓 + 30푔푡푓 + 42ℎ푡푓 , 푡 = 푡푓 푠푒푐 III. ROBOT TRAJECTORY PLANNING (23) (23) As discussed earlier, robot trajectory planning is to ensure the smooth variation in the robotic joints. Trajectory 휃(푡)⃛ ⃛ planning also gives time history of position, velocity and 휃푖 = 6푑, 푡 = 0 푠푒푐 { = ⃛ 2 3 4 acceleration at the intermediate point as well as final and 휃푓 = 6푑 + 24푒푡푓 + 60푓푡푓 + 120푔푡푓 + 210ℎ푡푓, 푡 = 푡푓 푠푒푐 starting point [13]. (24) In the present case all the initial and final values of all a  00000001    the joints are known. Apart from that angular velocity at   b 2 3 4 5 6 7   i   ff f f f f ttttttt1 f  the beginning and at the end is kept zero [14].  is the angle c =    00000010     f      in degree whereas ti and tf are the initial and final time 2 3 4 5 6 d i  ff f f f t7t6t5t4t3t210 f      e f  respectively. i and f are the initial and final angles of the 00000200        i 2 3 4 5 f   joints. All the angles are in degree and θ , θ̇ θ̈ , θ⃛ and  t6200 f 12tf 20tf 30tf 42tf     i i i i g  f         푑 θ⃛ 00006000 i i   h   are the initial angular displacement, angular velocity,  2 3 4      푑푡  6000 24t 60tff 120tf 210tf   f  (25) angular acceleration, jerk and jounce respectively, ⃛ 푑 θf ̇ ̈ ⃛ ̇ ̈ ⃛ 푎 = 휃푖 푏 = 휃푖 푐 = 휃푖/2 푑 = 휃푖/6 (26) whereas θf, θf, θf a θf and are the final angular 푑푡 displacement, angular velocity, angular acceleration, jerk (휃 −휃 ) (0.5휃̈ −휃̈ ) ⃛ 푓 푖 5 ̇ ̇ 푓 푖 휃푖 th 푒 = 35 4 − 3 (3휃푓 + 4휃푖)+5 2 - (27) and jounce (4 time derivative) respectively [10]. 푡푓 푡푓 푡푓 6푡푓

derivative represents acceleration, whereas third and fourth ̈ ̈ ⃛ ̈ (휃푖−휃푓) 1 ̇ ̇ (10휃푖−7휃푓) 휃푖+0.5휃푓 푓 = 84 5 + 4 (39휃푓 + 45휃푖)+ 3 + 2 (28) derivative represents jerk and jounce respectively. All the 푡푓 푡푓 푡푓 2푡푓 initial and final conditions at time t=0 seconds and t=tf ̈ ̈ ⃛ ⃛ (휃푓−휃푖) 1 ̇ ̇ (13휃푓−15휃푖) 2휃푖 휃푓 (29) (final time) are shown in (38) to (42) respectively. All eight 푔 = 70 6 − 5 (34휃푓 + 36휃푖)+ 4 – 3 − 3 푡푓 푡푓 2푡푓 3푡푓 2푡푓 unknowns are found using AX=B (shown in 43). All the

initial conditions, initial angles, final angles and all ̈ ̈ ⃛ ⃛ (휃푖−휃푓) 10 ̇ ̇ (휃푖−휃푓) (휃푖+휃푓) coefficients are shown in Table V. In (27), the ℎ = 20 7 + 6 (휃푓 + 휃푖)+2 5 + 4 (30) 푡푓 푡푓 푡푓 6푡푓 displacement function with respect to time is shown for all the joints. Eq. 44 to 49 represents all unknown coefficients. (휃푓−휃푖) 4 (휃푖−휃푓) 5 (휃푓−휃푖) 6 (휃푖−휃푓) 7 휃(푡) = 휃푖 + 35 4 푡 + 84 5 푡 + 70 6 푡 + 20 7 푡 Eq. (50) represents generalized equation for displacement 푡푓 푡푓 푡푓 푡푓 with all initial and final conditions. Coefficient values are (31) shown in Table VI for nonic function for all the joints.

휃(푡) 휃(푡) = 푎 + 푏푡 + 푐푡2 + 푑푡3 + 푒푡4 + 푓푡5 + 푔푡6 + ℎ푡7 + 푖푡8 +푗푡9 −12.5 − 0.33푡4 + 0.13푡5 − 0.018푡6 + 0.0009푡7, 퐹표푟 푗표푖푛푡 1 (33)

120 − 0.03푡4 + 0.47푡5 − 0.066푡6 + 0.003푡7, 퐹표푟 푗표푖푛푡 2

4 5 6 7 = −1.86푡 + 0.74푡 − 0.10푡 + 0.005푡 , 퐹표푟 푗표푖푛푡 3 4 5 6 7 −0.16푡 + 0.06푡 − 0.009푡 + 0.0004푡 , 퐹표푟 푗표푖푛푡 4 휃(푡)̇ = 푏 + 2푐푡 + 3푑푡2 + 4푒푡3 + 5푓푡4 + 6푔푡5 + 7ℎ푡6 + 8푖푡7 + 9푗푡8 { −101 + 2.72푡4 − 1.09푡5 + 0.15푡6 − 0.0072푡7, 퐹표푟 푗표푖푛푡 5 (34) (32)

̈ 2 3 4 5 6 휃(푡) = 2푐 + 6푑푡 + 12푒푡 + 20푓푡 + 30푔푡 + 42ℎ푡 + 56푖푡 + TABLE IV. COEFFICIENTS VALUES FOR SEPTIC FUNCTION 72푗푡7 (35) Coefficients and Initial Joint 1 Joint 2 Joint 3 Joint 4 Joint 5 2 3 4 5 6 Conditions 휃(⃛푡) = 6푑 + 24푒푡 + 60푓푡 + 120푔푡 + 210ℎ푡 + 336푖푡 + 504푗푡

a -12.5 120 0.5 0.5 -101 (36)

b 0 0 0 0 0 푑 휃(푡)⃛ = 24푒 + 120푓푡 + 360푔푡2 + 840ℎ푡3 + 1680푖푡4 + 3024푗푡5 푑푡

c 0 0 0 0 0 (37)

d 0 0 0 0 0 휃(푡) = 휃푖 = 푎, 푡 = 0 푠푒푐 휃 = e -0.33 -0.03 -1.86 -0.16 2.72 { 푓 푎 + 푏푡 + 푐푡2 + 푑푡3 + 푒푡4 + 푓푡5 + 푔푡6 + ℎ푡7 + 푖푡 8 + 푗푡 9 푡 = 푡 푓 푓 푓 푓 푓 푓 푓 푓 푓 푓 f 0.13 0.47 0.74 0.06 -1.09 (38) g -0.018 -0.066 -0.1 -0.009 0.15 ̇ 0.0000 휃(푡) = h 0.003 0.005 0.0004 -0.0072 ̇ 9 휃푖 = 푏, 푡 = 0 푠푒푐 ̇ { 휃푓 = i -12.5 120 0.5 0.5 -101 2 3 4 5 6 7 8 푏 + 2푐푡푓 + 3푑푡푓 + 4푒푡푓 + 5푓푡푓 + 6푔푡푓 + 7ℎ푡푓 + 8푖푡푓 + 9푗푡푓 , 푡푓 푠푒푐

f 0.5 76 -69 -6 0.5 (39)

휃(푡̈ ) = TABLE V. INITIAL CODITIONS FOR ALL THE JOINTS ̈ 휃푖 = 2푐 푡 = 0푠푒푐 ̈ Joi nt 1 Joint 2 Joint 3 Joint 4 Joint 5 휃푓 = ̇ 2푐 + 6푑푡 + 12푒푡2 + 20푓푡3 + 30푔푡4 + 42ℎ푡4 + 56푖푡 6 + 72푗푡 7 휽( 풕) 0 0 0 0 0 푓 푓 푓 푓 푓 푓 푓 { 푎푡 푡 = 푡푓 푠푒푐 ̈ 0 0 0 0 0 휽(풕) 휽(⃛풕) 0 0 0 0 0 (40) 풅 휽(⃛풕 ) 0 0 0 0 0 풅풕 ⃛ 휃(푡) = 휃⃛ = 6푑 푡 = 0푠푒푐 B. Trajectory Planning Using 9th Order Polynomial 푖 ⃛ 휃푓 = Angular displacement with respect to time with the 6푑 + 24푒푡 + 60푓푡2 + 120푔푡3 + 210ℎ푡4 + 336푖푡 5 + 504푗푡 6, order of seven is shown in (33). It’s first, to fourth time 푓 푓 푓 푓 푓 푓 { 푎푡 푡 = 푡푓 derivatives are given in (34) to (37) respectively. Here the first derivative represents angular velocity, second time (41)

푑 휃(⃛푡) = 푑푡 TABLE VI. COEFFICIENTS VALUES FOR NONIC FUNCTION 푑 ⃛ 휃푖 = 24푒, 푎푡 푡 = 0 푠푒푐 Coefficients 푑푡 푑 and Initial Joint 1 Joint 2 Joint 3 Joint 4 Joint 5 ⃛ 휃푓 = Conditions 푑푡 2 3 4 5 6 a -12.5 120 0.5 0.5 -101 6푑 + 24푒푡푓 + 60푓푡푓 + 120푔푡푓 + 210ℎ푡푓 + 336푖푡푓 + 504푗푡푓 { 푎푡 푡 = 푡 푠푒푐 푓 b 0 0 0 0 0 (42) c 0 0 0 0 0

 i  d 0 0 0 0 0   a  f   0000000001     e 0 0 0 0 0 b  i   2 3 4 5 6 7 8 ttttttttt1 9    =    ff f f f f f f f  c   0000000010     f  f 0.2 -0.71 0.20 -0.1 1.63   d    2 3 4 5 6 7 8   i  ff f f f f f t9t8t7t6t5t4t3t210 f    e   0000000200     f  g -0.11 0.39 -0.11 0.054 -0.91  2 3 4 5 6 7  f     t6200 12t 20t 30t 42t 56t 72t    i ff f f f f f g    0000006000   h 0.024 -0.084 0.024 -0.011 0.19     f  2 3 4 5 6 h  0000 24t 60t ff 120t f 210t f 336t f 504t f        i   d   i -0.0023 0.0082 -0.0023 0.0011 -0.019 0000 24 00000 i     dt   2 3 4 5   j   0000 12024 t f 360t f 840t f 1680t f 3024t f    d  j 0.00008 0.00008 -0.0005 -0.00004 0.0007  f  dt  (43) i -12.5 120 0.5 0.5 -101

f 0.5 76 -69 -6 0.5

̈ ⃛ ⃛ ̇ 휃푖 휃푖 푑 휃푖 푎 = 휃푖 푏 = 휃푖 푐 = 푑 = 푒 = ( ) (44) 2 6 푑푡 24 IV. RESULTS AND DISCUSSION

TABLE VII. VALUES OF VELOCITY AND ACCELERATION AT THE INITIAL, ̇ ̇ ̈ ̈ ⃛ ̈ ⃛ MIDDLE AND FINAL POINT OF THE TRAJECTORY (휃푓−휃푖) (56휃푓+70휃푖) (21휃푖−35휃푓) (휃푓+2.5휃푖) 푑 휃푓 푓 = 126 5 − 4 + 3 - 2 + ( ) (45) 푡푓 푡푓 2푡푓 푡푓 푑푡 푡푓 Order of Function 0 sec 3 sec 6 sec Velocity 푑 푑 In 0 17 33 (휃 −휃 ) (49휃̇ +61휃̇ ) (105휃̈ −77휃̈ ) (20휃⃛ +11.5휃⃛ ) 2.5 휃⃛ − 휃⃛ 푖 푓 푓 푖 푖 푓 푖 푓 푑푡 푖 푑푡 푓 deg/sec 푔 = 420 6 + 4 5 + 4 + 3 + 2 Parabolic 푡푓 푡푓 2푡푓 3푡푓 6푡푓 Acc (46) In 5.5 5.5 5.5 Deg/sec^2 푑 5 푑 ̇ ̇ ̈ ̈ ⃛ ⃛ 휃⃛ − 휃⃛ (휃푓−휃푖) (13휃푓+14휃푖) (53휃푓−63휃푖) (15휃푖+11휃푓) 푑푡 푓 3푑푡 푖 Velocity 0 25 0 ℎ = 540 7 − 20 6 + 5 - 4 + 3 푡푓 푡푓 푡푓 2푡푓 4푡푓 Cubic (47) Acc 18 0 -17

Velocity 0 37 0 Septic 푑 5 푑 (휃 −휃 ) (휃̇ −40휃̇ ) (7휃̈ −7.5휃̈ ) (2휃⃛ +7휃⃛ ) 휃⃛ − 휃⃛ Acc 0 0 0 푖 푓 푓 푖 푖 푓 푖 푓 푑푡 푓 4푑푡 푖 푖 = 315 8 + 5 7 +5 6 + 5 − 4 푡푓 푡푓 푡푓 2푡푓 6푡푓 (48) Velocity 0 41 0 Nonic Acc 0 0 0 푑 푑 ̇ ̇ ̈ ̈ ⃛ ⃛ 휃⃛ − 휃⃛ (휃푓−휃푖) (휃푓+휃푖) (휃푓−휃푖) (휃푖+휃푓) 푑푡 푓 푑푡 푖 푗 = 70 9 − 35 8 +7.5 7 -5 6 − 4 푡푓 푡푓 푡푓 6푡푓 24푡푓 Table VI shows the values of velocity and acceleration (49) at the beginning, middle and end of the trajectory. Angular

(휃푓−휃푖) (휃푖−휃푓) (휃푓−휃푖) displacement, velocity and acceleration are compared and ( ) 5 6 7 휃 푡 = 휃푖 + 126 5 푡 + 420 6 푡 + 540 7 푡 + 푡푓 푡푓 푡푓 plotted in fig 5 to 7 respectively. However methodology for (휃푖−휃푓) 8 (휃푓−휃푖) 9 315 8 푡 + 70 9 푡 (50) parabolic function and cubic function are not shown in this 푡푓 푡푓 paper. Based on these plots, values of velocity and acceleration are mentioned in Table VI. Trajectory design 휃(푡) using second order or parabolic function provides zero −12.5 + 0.20푡5 − 0.11푡6 + 0.024푡7 − 0.0023푡8 + 0.00008푡9, 퐹표푟 퐽1 velocity at the beginning, but fails to provide zero velocity 120 − 0.71푡5 + 0.39푡6 − 0.084푡7 + 0.0082푡8 + 0.00008푡9, 퐹표푟 퐽2 = 0.20푡5 − 0.11푡6 + 0.024푡7 − 0.0023푡8 − 0.0005푡9, 퐹표푟 퐽3 at the end. It also provides constant acceleration through 5 6 7 8 9 the entire trajectory. Cubic polynomial is able to satisfy −0.1푡 + 0.054푡 − 0.011푡 + 0.0011푡 − 0.00004푡 , 퐹표푟 퐽4 { −101 + 1.63푡5 − 0.91푡6 + 0.19푡7 − 0.019푡8 + 0.0007푡9, 퐹표푟 퐽5 zero velocity at the ends, but fails to give zero acceleration for the same. Meanwhile higher order polynomials are able (51) to satisfy zero velocity and acceleration at the ends. However higher order functions leave over shoot acceleration and velocity at the middle of the trajectory.

V. CONCLUSIONS

From the work presented in this paper, it is concluded that the trajectory planning using septic and nonic functions is successfully deployed for maintaining the continuity of joint velocity and acceleration. It also satisfies zero velocity and acceleration at the ends of the trajectory. Septic and nonic functions give smoother trajectory compare to parabolic and cubic polynomials, which helps robot in delivering the food without wasting it. It also increases the battery life and life of a servo actuator due to its continuous response of velocity and acceleration. The limitation of the paper is that, higher order polynomials provide peaks of acceleration and velocity in Figure 5. Angular displacement comparision for Joint 5 the middle of the trajectory, which is not desired for

multiple point trajectory. In the future, trajectory planning can be done using optimization algorithms like PSO, GA and ANN. Also fuzzy PID based dynamic control system can be designed to have a more control over joint angles, velocity and acceleration.

CONFLICT OF INTEREST The authors declare no conflict of interest in this research.

AUTHOR CONTRIBUTIONS Please state each author's contribution to this work. Figure 6. Angular velocity comparision of Joint 5 Priyam Parikh did the trajectory planning using lower and higher order polynomials. He also developed the entire robotic arm along with programming in MATLAB and ARDUINO. Co-author Dr. Reena Trivedi did the motion planning, simulation and 3D modeling. Dr. Jatin Dave helped in calculating forward and inverse kinematics of the serial manipulator.

REFERENCES

[1] A. Valente, S. Baraldo, and E. Carpanzano, “Smooth trajectory generation for industrial robots performing high precision assembly processes,” CIRP Annals, vol. 66, no. 1, pp. 17–20, 2017. [2] A. Gasparetto and V. Zanotto, “Optimal trajectory planning for industrial robots, ” Advances in Engineering Software, vol. 41, no. 4, pp. 548–556, 2010. Figure 7. Angular acceleration comparision of Joint 5 [3] W. G. Hao, Y. Y. Leck, L. C. Hun, “ 6-DOF PC-Based Robotic Arm (PC-ROBOARM) with efficient trajectory planning and speed control,” in Proc. 2011 4th International Conference on Mechatronics (ICOM) , 2011. [4] F. Basile, F. Caccavale, P. Chiacchio, J. Coppola, C. Curatella, “Task-oriented motion planning for multi-arm robotic systems,” Robotics and Computer-Integrated Manufacturing, vol. 28, no. 5, pp. 569–582, 2012. [5] S. Fang, X. Ma, J. Qu, S. Zhang, N. Lu, X. Zhao, “Trajectory planning for seven-DOF robotic arm based on seventh degree polynomial,” in: Jia Y., Du J., Zhang W. (eds) Proceedings of 2019 Chinese Intelligent Systems Conference. CISC 2019. Lecture Notes in Electrical Engineering, vol. 593. Springer, Singapore, 2020. [6] J. Zhang, Q. Meng, X. Feng, H. Shen, “A 6-DOF robot-time optimal trajectory planning based on an improved genetic algorithm,” Robotics and Biomimetics, vol. 5, no. 1, 2018 [7] Y. Li, T. Huang, D. G. Chetwynd, “An approach for smooth trajectory planning of high-speed pick-and-place parallel robots using quintic B-splines,” Mechanism and Machine Theory, vol. 126, - pp. 479–490. th Figure 8. Final Postion of the Robot in XZ plane after applying 7 order polynomial method (real time)

[8] X. J. Zhao, M. L. Wang, N. Liu, and Y. W. Tang, “Trajectory Copyright © 2020 by the authors. This is an open access article planning for 6-DOF robotic arm based on quintic polynomial,” in distributed under the Creative Commons Attribution License (CC Proc. the 2017 2nd International Conference on Control, BY-NC-ND 4.0), which permits use, distribution and reproduction in any Automation and Artificial Intelligence (CAAI 2017), [Online]. medium, provided that the article is properly cited, the use is Available: https://doi.org/10.2991/caai-17.2017.23 non-commercial and no modifications or adaptations are made. [9] J. Rosell, A. Perez, A. Aliakbar, Muhayyuddin, L. Palomo, and N. Garcia, “The Kautham project: A teaching and research tool for robot motion planning,” in Proc. the 2014 IEEE Emerging Technology and Factory Automation (ETFA), 2014. Priyam Parikh is currently pursuing his PhD [10] J. M. Longval, C. Gosselin, “Dynamic trajectory planning and in mechanical engineering from Nirma geometric design of a two-DOF translational cable-suspended University, Gujarat, India. He is also working planar parallel robot using a parallelogram cable loop,” vol. 5B, as an Assistant Professor at Anant National 42nd Mechanisms and Robotics Conference, 2018. University in Industrial Design Department. He [11] X. Jiang, E. Barnett, and C. Gosselin, “Dynamic point-to-point has done his masters in Mechatronics trajectory planning beyond the static workspace for six-DOF Engineering from G.H. Patel College of cable-suspended parallel robots,” IEEE Transactions on Robotics, Engineering and technology, VVnagar, India. vol. 34, no. 3, pp. 781–793, 2018. His area of interest is Robotics, Automation, [12] X. Wang, D. Zhang, C. Zhao, H. Zhang, and H. Yan, ”Singularity Mechatronics and Industrial Design. analysis and treatment for a 7R 6-DOF painting robot with non-spherical wrist,” Mechanism and Machine Theory, vol. 126, pp. 92–107, 2018. [13] M. Vulliez, S. Zeghloul, and O. Khatib, “Kinematic analysis of the delthaptic, a new 6-DOF haptic device,” Springer Proceedings in Advanced Robotics, pp. 181–189, 2017. [14] S. Zhu, Wang, “Time-optimal and jerk-continuous trajectory planning algorithm for manipulators,” J Mech Eng., 2010;46(3):456–62. [Online]. Available: https://doi.org/10.3901/jme.20.