0
Research Papers

Analysis of Optimal Balancing for Robotic Manipulators in Repetitive MotionsOPEN ACCESS

[+] Author and Article Information
Amin Nikoobin

Robotics and Control Lab,
Faculty of Mechanical Engineering,
Semnan University,
Semnan 3513119111, Iran
e-mail: anikoobin@semnan.ac.ir

Faculty of Computer and Electrical Engineering,
Semnan University,
Semnan 3513119111, Iran

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received January 18, 2017; final manuscript received December 13, 2017; published online March 7, 2018. Assoc. Editor: Yunjun Xu.

J. Dyn. Sys., Meas., Control 140(8), 081002 (Mar 07, 2018) (8 pages) Paper No: DS-17-1034; doi: 10.1115/1.4039155 History: Received January 18, 2017; Revised December 13, 2017

Abstract

Balancing plays a major role in performance improvement of robotic manipulators. From an optimization point of view, some balancing parameters can be modified to decrease motion cost. Recently introduced, this concept is called optimal balancing: an umbrella term for static balancing and other balancing methods. In this method, the best combination of balancing and trajectory planning is sought. In this note, repetitive full cycle motion of robot manipulators including different subtasks is considered. The basic idea arises from the fact that, upon changing dynamic equations of a robotic manipulator or cost functions in subtasks, the entire cycle of motion must be reconsidered in an optimal balancing problem. The possibility of cost reduction for a closed contour in potential fields is shown by some simulations done for a PUMA-like robot. Also, the obtained results show 34.8% cost reduction compared to that of static balancing.

<>

Introduction

In many industries, robots are increasingly being used in repetitive tasks. They should be properly programmed to do many monotonous motions such as point-to-point (PTP) and/or path tracking actions. When an end effector of a manipulator attains an initial and a final pose, the manipulator is said to be employed for a PTP action. A major category of PTP motions is comprised of pick-and-place operations such as changing tools in machine tools, loading/unloading belt conveyors, and such simple assembly operations as putting roller bearings on a shaft. Path tracking actions appear in many operations such as gluing, flame-cutting, routing, arc-welding, and deburring.

Different from the simple motion planning problems as mentioned above, in manufacturing industry, there exist two other classes of complex tasks called multipoints manufacturing and cyclic tasks. Multipoints manufacturing tasks have many unordered points and hence it is necessary to plan an optimal strategy to traverse all the desired points in an orderly way while satisfying the required performance index [1]. On the other hand, cyclic robotic tasks can be decomposed into subtasks such as picking, moving, and placing. Conventionally, the subtasks are independent and their optimality conditions can be applied independently. For instance, in conventional trajectory planning for a pick-and-place task, forward and return phases are planned completely apart [2]. Nevertheless, if designing a common parameter all along the motion is desired, the independency does not hold anymore and all the subtasks must be considered in a one-shot optimization, simultaneously. In this method, a parametric trajectory optimization is presented for simultaneously computing physical parameters, actuation requirements, and robot motions for more efficient robot designs [3].

Optimization of cost with respect to dynamic equations of a robot is known as common optimal control problem (OCP). This problem can be solved using either of the two major methods, namely indirect and direct methods [4,5]. The direct methods look for optimal values regardless of optimality conditions, while the indirect methods are based on optimality condition solution. In spite of global convergence and constraint applicability of direct methods, they are not as accurate as indirect methods. In the indirect method, the Pontryagin's minimum principle transforms an OCP into a multipoint boundary value problem whose solution is the optimal trajectory [68]. The optimality conditions and transversality conditions are extracted in indirect approach only. Note that with an appropriate initial guess, indirect approach has a rapid convergence. Nevertheless, this is not possible for all the problems primarily and this benefits direct solution. In general, the indirect method is superior to the direct method in accuracy and convergence speed. However, it is difficult for the indirect method to handle the problems subject to input and state constraints [9,10].

Balancing is another efficient way to improve the performance of robot motions. In literature, it is extensively considered as a modification and/or addition of passive elements such as links, counterweights [11], or springs [12], and is divided into two major types: complete and partial balancing [13]. In complete balancing, all the gravitational, velocity-based coupling terms and inertial couplings are released [14], which results in heavy counterweights and counter rotors that make balancing impossible for usual applications. On the other hand, a few researchers have defined a performance index for partial balancing problem [15] based on some predefined trajectories or some dynamical properties. Since the choice of balancing parameters changes dynamics and optimal trajectory of the system, the balancing parameters, and optimal trajectories cannot be designed individually. This hint motivates the idea of optimal balancing method (OBM) proposed by Nikoobin and Moradi [16] as the most general mathematical theory about partial balancing. Moreover, it has been shown that the static balancing is a special case of OBM [17]. In OBM, some unknown parameters are found simultaneously with trajectory unknowns such as position waypoints and velocity profiles. In the previous studies on the application of OBM for balancing [16,17], only one stage of motion was considered, making it hard to make a decision about overall performance improvement. For instance, the values of payload for carrying and returning phases are different. So, applying OBM to a carrying phase is not necessarily the optimal choice for the overall motion.

The major contribution of this work is to consider OBM for repetitive tasks. Furthermore, some simplifications are applied on the previous formulations [16] and then extended to general multitask motions. In order to evaluate the proposed method, it is applied on a PUMA-like robot. The simulations indicate that although the dominant static balancing method shows a lot of improvements in performance, it can be improved further by OBM. Moreover, the obtained results from OBM can be used as a point of reference for performance evaluation of any balancing approach.

Optimal Balancing for Mechanical Systems

State Space Representation.

Dynamical model of an open-chain, full-actuated mechanical system can be described as a lumped model system with n degrees-of-freedom (DOFs) Display Formula

(1)$M(q)q¨+C(q,q˙)+G(q)=u$

where $u∈Rn$ is the vector of input torques, $q,q˙$, and $q¨∈Rn$ are the vectors of joint angular positions, velocities, and accelerations, respectively, $M(q)∈Rn×n$ is the inertia matrix, $C(q,q˙)∈Rn$ represents the centripetal and the Coriolis forces, and $G(q)∈Rn$ describes the gravity effects. By defining the continuous state vector as Display Formula

(2)$x=[x1Tx2T]T=[qTq˙T]T∈R2n$
the dynamics, Eq. (1) can be rewritten in the state space form as Display Formula
(3)$x˙=[x2f(x1,x2)]=[x2M−1(x1)[u−C(x1,x2)−G(x1)]]$

The One Stage Optimal Balancing Method Statement.

Optimal balancing method is an OCP that calculates optimal unknown parameters and trajectory [16]. The OCP can be stated as: Find a continuous admissible control history $u[t0tf]→Ω⊆Rn$ (where $Ω$ is an acceptable region in $Rm$) and a vector of parameters $b∈Rnp$, generating the corresponding state trajectory $x[t0tf]→R2n$, which minimizes a quadratic in the control cost function defined as Display Formula

(4)$J=ϕ(x0,xf,b)+∫t0tf[uTRu+L(x,b)]dt$
subject to the system dynamics (3) and the given initial condition Display Formula
(5)$x(t0)=x0$

and the prescribed final conditions Display Formula

(6)$x(tf)=xf$

Here, $t0$ and $tf$ are the initial time and final time, respectively, $x0$ and $xf$ are the predefined initial and final state vectors, respectively, $ϕ$ and $L(x,b)$ are scalar continuously differentiable functions and R is weighing input matrix. The vector of unknown parameters b is the vector of robot parameters such as counterweights must be determined. Note that b is constant in the motion not in the design stage. The vector of unknown parameters b can be simply considered as constant states. This meant some zero state equations should be appended to the state space dealing with the constant parameters. These additional states can be considered as follows and the appended state space becomes: Display Formula

(7)$x˙′=[x˙b˙]$

where $b˙=0$. The co-state vector can be defined as a concatenation of the three co-state vectors $λ1∈Rn$ as co-positions, $λ2∈Rn$ as co-velocities, and $λb∈Rnp$ as co-parameters, and then the Hamiltonian function can be constructed as Display Formula

(8)$H=uTRu+L+[λ1Tλ2TλbT]︸λT[x2f0]︸x˙′=L+λ1Tx2+λ2Tf$

where $λ∈R2n+np$ is the co-state vector of the system. Application of the PMP leads the problem into a two-point BVP as a complete set of 4n + np ordinary differential equations (ODE) [18], including the 2n ODEs for the states given by Eq. (3) as Display Formula

(9)$x˙1=Hλ1=x2x˙2=Hλ2=M−1(x1)[u−C(x1,x2)−G(x1)]$

and 2n + np ODEs for the co-states derived as Display Formula

(10)$λ˙1,i=−Hx1,iT=−∂L∂x1,iT+M−1∂M∂x1,ifλ2+M−1(∂C∂x1,i+∂G∂x1,i)Tλ2, i∈[1…n]λ˙2,i=−Hx2,iT=−∂L∂x2,iT−λ1+M−1∂C∂x2,iTλ2, i∈[1…n]λ˙bi=−HbiT=−∂L∂biT+M−1∂M∂bifλ2+M−1(∂C∂bi+∂G∂bi)Tλ2, i∈[1…np]$

and with n algebraic equations as Display Formula

(11)$Hu=0⇒u=−R−1M−1λ2$

and 4n + 2np BCs as Display Formula

(12)$[x1x2λ1λ2λb]={[x1,0x2,0freefree0]@(t=t0);[x1,fx2,ffreefreeφb]@(t=tf)}$

These equations can be solved by using BVP solvers such as the bvp4c code implemented in the MATLAB by Kierzenka and Shampine [19].

Repetitive motion is usual in industrial settings, e.g., assembly lines. It should be pointed out that, for repetitive trajectories, the above-mentioned compensating contributions can be computed offline and properly stored on the basis of a tradeoff clarification between memory capacity and computational requirements of any control architecture.

Consider a robot motion consisting of m subtasks, with the ith subtask being associated with a desired cost function $Ji$ and a dynamic equation $fi$ in a time interval [ti-1ti], as shown in Fig. 1. Because of picking and detachment of some payload masses in pick-and-place tasks, the dynamic equations can be changed in the subtasks.

Two common examples of repeated tasks are shown in Fig. 2, including material handling process and constant velocity gluing. A pick-and-place process is first shown for handling some materials from a cluttered source and putting them into boxes to be conveyed. Also, a gluing process is shown subsequently; the process is composed of two major actions: a semicircular gluing and a PTP motion from the final position to the initial one. Details of the tasks are given in Table 1.

As it can be seen from the previous examples, three major subtasks can be defined, free motion, path tracking, and halt. So, the cost function for the ith subtask where 1 ≤ im ranges within [ti ti+1] can be stated as follows: Display Formula

(13)$minJi=ζiφi+∫ti−1ti(uTRiu+γiLi)dt s.t. x˙=[x2fi]$

where $φi$ is the halt cost and $Li$ is the cost of the path tracking error. $ζi$, $Ri$, and $γi$ are the penalty terms of the corresponding subtasks. For the halt subtask, $Ri=γi=0$; for the free motion subtask, $ζi=γi=0$; and for the path tracking subtask, $ζi=0$, where $Ri$ and $γi$ must be justified depending on the tracking error.

When the robot is not fully balanced, in order to retain the robot in a fixed position, some energy needs to be consumed. In halt subtask, the applied torque on joints is obtained as Display Formula

(14)$τpayload,i=JiTFei$

where $JiT$ is the Jacobian matrix of the manipulator and $Fei$ is the general exerted forces on the end effector defined as Display Formula

(15)$Fei=[feiMei]$

where $fei$ and $Mei$ are resultant of the exerted forces and torques on the end effector, respectively. This torque should be added to gravitational terms. Total torque can be expressed as Display Formula

(16)$τtotal,i=τpayload,i+τstatic,i=([00−g0mpayload,i000]×Ji)T+G(qi)$

So, the halt cost for a td,i-second delay can be stated as Display Formula

(17)$ϕi=τtotal,iTRiτtotal,i×td,i$

Trying to include multiple tasks in a single cost function and dynamics and express the result in a unified and abstract manner, a Heaviside-like function $hi$ is defined here (Fig. 3) Display Formula

(18)$hi={1ti−1≤t≤ti0else$

The overall cost function can be expressed as Display Formula

(19)$minJ=∑i=1mhiJi s.t. x˙=∑i=1mhi[x2fi]$

Finally, by substituting Eq. (13) into Eq. (19), one can write Display Formula

(20)$minJ=∑i=1mζiϕi+∫t0tf∑i=1m[hi(uTRiu+γiLi)dt] s.t. x˙=∑i=1mhi[x2fi]$

The derivations in middle times ti are not defined since the Heaviside function is not continuous at such point. But, this discontinuity is allowed since it is only occurred at boundaries.

Optimal Control Problem Formulation With the Free Middle-Times.

The time ti in the subtasks is unknown in general. Therefore, the problem concerned here is a multipoint BVP with at least m − 1 unknown parameters for middle-time parameters along with r unknown design parameters. Also, in some cases, the final time tf may be free. Therefore, time-scales for all of the stages are defined as follows: Display Formula

(21)$τi=t−ti−1ti−ti−1,i=1…m$

where the boundaries are arranged in the integer sequence $τ=[12⋯m]$. So, the differential relationship between real time and scaled time can be stated as follows: Display Formula

(22)$dt=(ti−ti−1)dτi$

Now, the sub-OCP (13) becomes Display Formula

(23)$minJi=(ti−ti−1)∫i−1i(uTRiu+γiLi)dτi s.t. x˙=(ti−ti−1)hi[x2fi]$

and the overall OCP (20) becomes Display Formula

(24)$minJ=∑i=1mϕi+∫0m∑i=1m−1[hi′(ti−ti−1)(uTRiu+γiLi)dτi] s.t. x˙=∑i=1m(hi′(ti−ti−1)[x2fi])$

where $hi′$ is an integer Heaviside-like function defined as Display Formula

(25)$hi′={1i−1≤τ≤i0else$

The Hamiltonian is constructed by Eq. (24) as follows: Display Formula

(26)$H=∑i=1mhi′(ti−ti−1)(uTRiu+γiLi+λ1Tx2+λ2Tf1)$

Therefore, the costate equations can be derived as follows: Display Formula

(27)$λ˙1=−∇x1HT=−∑i=1mhi(ti−ti−1)(γi∂Li∂x1T+∇x1fiλ2)λ˙2=−∇x2HT=−∑i=1mhi(ti−ti−1)(γi∂Li∂x2T+λ1+Mi−1∂Ci∂x2Tλ2)$
Display Formula
(28)$λ˙bj=−∂H∂bjT=−∑i=1mhi(ti−ti−1)(γi∂Li∂bjT+∇bjfiλ2), forj=1…np$

where $∇afi=Mi−1∂Mi/∂afi+Mi−1(∂Ci/∂a+∂Gi/∂a)T$ for a = x1 or bj.

The value of ti is unknown and can be simply considered as a constant state in the trajectory. Therefore, an additional scalar costate $λti$ is defined for the unknown middle times (ti). The optimality condition with respect to the middle times can be extracted as follows: Display Formula

(29)$λ˙mf=−∂H∂tf,λ˙ti=−∂H∂ti$

But here, for each middle time ti, there are two constraints as: ti-1ti and titi+1. In order to eliminate these constraints, unconstrained free parameters are defined as the following recursive equations: Display Formula

(30)$ti−1=1+ε+sin ηi2ti,tm=tf2$

where $ηi$ is a free variable and ε can be any positive value. Therefore, for t1 = 0 and tm = tf as a final time, it can be written that Display Formula

(31)$ti−1=tf2∏k=im1+ε+sin ηk2$

According to this definition, no constraint is required for the middle times and the final time. Now, the optimality condition with respect to ηi can be expressed as Display Formula

(32)$λ˙mj=−∂H∂ηjT=∑i=1m(−∂H∂ti−1∂ti−1∂ηj)$

where Display Formula

(33)$−∂H∂ti−1=hi−1(uTRi−1u+γi−1Li−1+λ1Tx2+λ2Tfi−1)−hi(uTRiu+γiLi+λ1Tx2+λ2Tfi)$
Display Formula
(34)$∂ti−1∂ηj={0j

For free final time boundary problems, the optimality condition with respect to tf can be expressed as Display Formula

(35)$∂ti−1∂tf=2ti−1tf$

then: Display Formula

(36)$λ˙mf=−∂H∂tfT=∑i=1m−∂H∂ti−1∂ti−1∂tf,=2 cos ηj1+ε+sin ηj∑i=1mti−1(hi−1(uTRi−1u+γi−1Li−1+λ1Tx2+λ2Tfi−1)−hi(uTRiu+γiLi+λ1Tx2+λ2Tfi))$

The optimal control can be rewritten as Display Formula

(37)$∇uH=0⇒u=−R−1Mi−1ti−ti−1λ2T for i−1<τ

The initial BCs (i = 0) are as follows: Display Formula

(38)$[x1x2λ1λ2λbλm]=[x1,0x2,0freefree00] @(τ=0)$

and for the ith middle time, we have Display Formula

(39)$[x1x2λ1λ2λbλm]=[x1,ix2,ifreefreefreefree] @(τ=i),i∈[1…m−1]$
while at the final time (i = s), we have Display Formula
(40)$[x1x2λ1λ2λbλm]=[x1,fx2,ffreefreeφb|τ=m0] @(τ=m)$

Additional middle point conditions should be considered (due to the continuity of the state vector in the middle boundaries) as Display Formula

(41)$x1/2[(i−1)+]=x1/2[i−]=x1/2(i)=x1/2,i$

where $xi$ are the given middle states (position or velocity) at $τ=i, i−1+$ denotes the right boundary of the i − 1th segment and i- denotes the left boundary of the ith segment. Notice that the costates are not continuous in the middle boundaries.

Finally, for the robot with n DOFs, 2n equations can be derived from Eq. (3). Then for m subtasks, 2 nm equations can be obtained from Eq. (13) with 2 nm costate equations from Eq. (27). Therefore, in order to optimize the trajectory, these 4 nm equations must be solved. For optimality of the parameters, np costates (27) are added to the problem, with m − 1 additional equations added from Eq. (32) for the optimality of middle times, making up a total of 4 mn + np + m − 1 equations to be solved. Finally, for the completeness of the BVP, a total of 4mn + 2np + 2m − 2 BCs are required. This can be supplied by 2n + np + m − 1 BCs from Eq. (38), 4(m − 1)n BCs from Eqs. (39) and (41), and 2n + np + m − 1 BCs from Eq. (40).

Case Study: Dynamic Model of a Puma-Like Robot

The PUMA-like robot studied here is a rigid serial robot of 3DOFs (Fig. 4) [20]. Its DH parameters are listed in Table 2, with the simplified characteristics of the robot dynamics given in Table 3. Elements of the position of the center of gravity are reported as rx, ry, and rz, referring to x, y, and z coordinates, respectively, in a coordinate frame close to the corresponding link. The diagonal terms of inertia tensor are represented by Ixx, Iyy, and Izz for each link, when the coordinate frame is fixed at the center of gravity. DH parameters of the robot are given in Table 2 and parameters of a link are stated in Table 3 [17]. Note that in many cases, the values of cross inertia are negligible as those have insignificant effects on overall optimal trajectory [21].

Here, the balancing is considered by two counterweights. For any pair of r1 and r2 and payload mp, there are corresponding mc1 and mc2 values, which may be different for static and optimal cases. Here, the values of r1 and r2 are considered as 0.25 and 0.125, respectively, and so, by eliminating the gravitational torques, the counterweights are calculated as mc1 = 40 kg and mc2 = 10 kg for the static balanced case of 2 kg payload. For the optimal balanced case, the counterweights should be computed through the OBM stated above and then applied.

Motion Definitions.

In PTP motion, initial and final positions are given without any path allocation. Therefore, instantaneous position and velocity profile should be computed. Using the inverse kinematics of robot, it is possible to find BCs at the joint coordinates.

In the PTP studied here, it is desired to pick a payload from an initial position and then place it in a desired final position. Subsequently, it returns without any payload to the initial position. Let the payload be 2 kg, the initial position of the end effector in Cartesian space at t = 0 is X0, and the final position at unknown ts is Xf. The trajectory should be applied rapidly within tf = 0.9 s with a zero velocity at boundaries. The values are stated as Display Formula

(42)$(X0=[0.60.5−0.1]m≡x1=[0.710.49−1.29]rad)t=0,tf→(Xs=[0.5−0.40.5]m≡xs=[2.411.751.36]rad)t=tsx2(0)=x2(ts)=x2(tf)=0$

The BCs are stated in Table 4.

Simulations.

In this section, three cases are compared to show advantages of OBM over static balancing. We call the unbalanced case “normal,” the “static” balanced case with no payload and, the OBM-based case, “optimal.” In the static balancing case, positions of the counterweights are considered at r1 = 0.25 m and r2 = 0.125 m.

The objective of the PTP task (within 0.9 s) is to pick a 2 kg mass from the position X0, to place it into the position Xf, and to return from the position Xf to the position X0. Note that the picking time is arbitrary and it is optimized in the simulation.

The resultant trajectories, velocities, and torques are shown in Figs. 57 (see details in Table 5). Finally, for the better comparison, the cost values are plotted in Fig. 8, which shows a considerable reduction in the cost function for the optimal balancing method. The effect of reduction in the cost function is directly reflected in the required maximum value of torques illustrated in Fig. 7.

Discussion

For this PTP motion, there is a payload in the forward pass, which is released on the way back. Therefore, different dynamic models hold in either trajectory. In this problem, the cost function of the static balancing method decreased by 94% compared to the normal case, while the cost function of the proposed method was just 34.8% of the static balancing cost. In Table 5, it is obvious that the duration of picking (first stage) and placing (second stage) can be different. In Fig. 5, it is observed that the path of the optimal case is smoother than that of other cases, and this may be an engineering sense for the optimized motion.

Conclusion

In this research, OBM was extended to repetitive motions, which has not been considered before. To this end, a general task including different subtasks was considered, in which each subtask could have its own cost function and dynamic equation. Then, by defining a general cost function, considering the middle times, and balancing parameters as unknowns, a compact formulation was derived from the indirect solution of the optimal control problem. In order to show the effectiveness of the proposed method, some simulations were run on a PUMA-like robot. The results indicated the superiority of optimal balancing over the static balancing approach. Notably, this study demonstrated the possibility of designing passive parameters that can assist the manipulator in decreasing the cost in a potentially conservative field for a closed path. An important future work can be the experimental validation of the optimal balancing. Also, it is significant to study the impact of constraints on the optimal balancing superiority and considering the minimum time problem to see the effect of optimal balancing in the cycle time. In this, work, only the mass of the counterweights is considered as the design parameter, while it is predicted that adding another adjustable parameter such as angle of counterweights leads to the wonderful result, “zero-power trajectories.”

Acknowledgements

• Iran National Science Foundation (INSF) (Grant No. 95839116).

References

Zhang, Q. , and Zhao, M.-Y. , 2016, “ Minimum Time Path Planning of Robotic Manipulator in Drilling/Spot Welding Tasks,” J. Comput. Des. Eng., 3(2), pp. 132–139.
Gosselin, C. , and Foucault, S. , 2014, “ Dynamic Point-to-Point Trajectory Planning of a Two-DOF Cable-Suspended Parallel Robot,” IEEE Trans. Rob., 30(3), pp. 728–736.
Spielberg, A. , Araki, B. , Sung, C. , Tedrake, R. , and Rus, D. , 2017, “ Functional Co-Optimization of Articulated Robots,” IEEE International Conference on Robotics and Automation (ICRA), Singapore, May 29–June 3, pp. 5035–5042.
von Stryk, O. , and Bulirsch, R. , 1992, “ Direct and Indirect Methods for Trajectory Optimization,” Ann. Oper. Res., 37(1), pp. 357–373.
Nikoobin, A. , and Moradi, M. , 2017, “ Indirect Solution of Optimal Control Problems With State Variable Inequality Constraints: Finite Difference Approximation,” Robotica, 35(1), pp. 50–72.
Gasparetto, A. , Lanzutti, A. , Vidoni, R. , and Zanotto, V. , 2011, “ Validation of Minimum Time-Jerk Algorithms for Trajectory Planning of Industrial Robots,” ASME J. Mech. Rob., 3(3), p. 031003.
Ghanaatpishe, M. , and Fathy, H. K. , 2017, “ On the Structure of the Optimal Solution to a Periodic Drug-Delivery Problem,” ASME J. Dyn. Syst., Meas., Control, 139(7), p. 071001.
Pappalardo, C. M. , and Guida, D. , 2017, “ Adjoint-Based Optimization Procedure for Active Vibration Control of Nonlinear Mechanical Systems,” ASME J. Dyn. Syst., Meas., Control, 139(8), p. 081010.
Wang, Z. , and Li, Y. , 2017, “ An Indirect Method for Inequality Constrained Optimal Control Problems,” IFAC PapersOnLine, 50(1), pp. 4070–4075.
Fabien, B. C. , 2013, “ Indirect Solution of Inequality Constrained and Singular Optimal Control Problems Via a Simple Continuation Method,” ASME J. Dyn. Syst., Meas., Control, 136(2), p. 021003.
Diken, H. , 1995, “ Effect of Mass Balancing on the Actuator Torques of a Manipulator,” Mech. Mach. Theory, 30(4), pp. 495–500.
Boisclair, J. , Richard, P.-L. , Laliberté, T. , and Gosselin, C. , 2017, “ Gravity Compensation of Robotic Manipulators Using Cylindrical Halbach Arrays,” IEEE/ASME Trans. Mechatronics, 22(1), pp. 457–464.
Arakelian, V. , 2017, “ Inertia Forces and Moments Balancing in Robot Manipulators: A Review,” Adv. Rob., 31(14), pp. 717–726.
Moradi, M. , Nikoobin, A. , and Azadi, S. , 2010, “ Adaptive Decoupling for Open Chain Planar Robots,” Sci. Iran. Trans. B, 17(5), pp. 376–386.
Herman, P. , 2008, “ Dynamical Couplings Reduction for Rigid Manipulators Using Generalized Velocity Components,” Mech. Res. Commun., 35(8), pp. 553–561.
Nikoobin, A. , and Moradi, M. , 2011, “ Optimal Balancing of Robot Manipulators in Point-to-Point Motion,” Robotica, 29(02), pp. 233–244.
Nikoobin, A. , Moradi, M. , and Esmaili, A. , 2013, “ Optimal Spring Balancing of Robot Manipulators in Point-to-Point Motion,” Robotica, 31(4), pp. 611–621.
Ansari, A. , and Murphey, T. , 2015, “ Minimum Sensitivity Control for Planning With Parametric and Hybrid Uncertainty,” Int. J. Rob. Res., 35(7), pp. 823–839.
Kierzenka, J. , and Shampine, L. F. , 2001, “ A BVP Solver Based on Residual Control and the Matlab PSE,” ACM Trans. Math. Software (TOMS), 27(3), pp. 299–316.
Kebria, P. M. , Abdi, H. , and Nahavandi, S. , 2016, “ Development and Evaluation of a Symbolic Modelling Tool for Serial Manipulators With Any Number of Degrees of Freedom,” IEEE International Conference on Systems, Man, and Cybernetics (SMC), Budapest, Hungary, Oct. 9–12, p. 004223.
Herman, P. , 2009, “ Dynamical Couplings Analysis of Rigid Manipulators,” Meccanica, 44(1), pp. 61–70.
View article in PDF format.

References

Zhang, Q. , and Zhao, M.-Y. , 2016, “ Minimum Time Path Planning of Robotic Manipulator in Drilling/Spot Welding Tasks,” J. Comput. Des. Eng., 3(2), pp. 132–139.
Gosselin, C. , and Foucault, S. , 2014, “ Dynamic Point-to-Point Trajectory Planning of a Two-DOF Cable-Suspended Parallel Robot,” IEEE Trans. Rob., 30(3), pp. 728–736.
Spielberg, A. , Araki, B. , Sung, C. , Tedrake, R. , and Rus, D. , 2017, “ Functional Co-Optimization of Articulated Robots,” IEEE International Conference on Robotics and Automation (ICRA), Singapore, May 29–June 3, pp. 5035–5042.
von Stryk, O. , and Bulirsch, R. , 1992, “ Direct and Indirect Methods for Trajectory Optimization,” Ann. Oper. Res., 37(1), pp. 357–373.
Nikoobin, A. , and Moradi, M. , 2017, “ Indirect Solution of Optimal Control Problems With State Variable Inequality Constraints: Finite Difference Approximation,” Robotica, 35(1), pp. 50–72.
Gasparetto, A. , Lanzutti, A. , Vidoni, R. , and Zanotto, V. , 2011, “ Validation of Minimum Time-Jerk Algorithms for Trajectory Planning of Industrial Robots,” ASME J. Mech. Rob., 3(3), p. 031003.
Ghanaatpishe, M. , and Fathy, H. K. , 2017, “ On the Structure of the Optimal Solution to a Periodic Drug-Delivery Problem,” ASME J. Dyn. Syst., Meas., Control, 139(7), p. 071001.
Pappalardo, C. M. , and Guida, D. , 2017, “ Adjoint-Based Optimization Procedure for Active Vibration Control of Nonlinear Mechanical Systems,” ASME J. Dyn. Syst., Meas., Control, 139(8), p. 081010.
Wang, Z. , and Li, Y. , 2017, “ An Indirect Method for Inequality Constrained Optimal Control Problems,” IFAC PapersOnLine, 50(1), pp. 4070–4075.
Fabien, B. C. , 2013, “ Indirect Solution of Inequality Constrained and Singular Optimal Control Problems Via a Simple Continuation Method,” ASME J. Dyn. Syst., Meas., Control, 136(2), p. 021003.
Diken, H. , 1995, “ Effect of Mass Balancing on the Actuator Torques of a Manipulator,” Mech. Mach. Theory, 30(4), pp. 495–500.
Boisclair, J. , Richard, P.-L. , Laliberté, T. , and Gosselin, C. , 2017, “ Gravity Compensation of Robotic Manipulators Using Cylindrical Halbach Arrays,” IEEE/ASME Trans. Mechatronics, 22(1), pp. 457–464.
Arakelian, V. , 2017, “ Inertia Forces and Moments Balancing in Robot Manipulators: A Review,” Adv. Rob., 31(14), pp. 717–726.
Moradi, M. , Nikoobin, A. , and Azadi, S. , 2010, “ Adaptive Decoupling for Open Chain Planar Robots,” Sci. Iran. Trans. B, 17(5), pp. 376–386.
Herman, P. , 2008, “ Dynamical Couplings Reduction for Rigid Manipulators Using Generalized Velocity Components,” Mech. Res. Commun., 35(8), pp. 553–561.
Nikoobin, A. , and Moradi, M. , 2011, “ Optimal Balancing of Robot Manipulators in Point-to-Point Motion,” Robotica, 29(02), pp. 233–244.
Nikoobin, A. , Moradi, M. , and Esmaili, A. , 2013, “ Optimal Spring Balancing of Robot Manipulators in Point-to-Point Motion,” Robotica, 31(4), pp. 611–621.
Ansari, A. , and Murphey, T. , 2015, “ Minimum Sensitivity Control for Planning With Parametric and Hybrid Uncertainty,” Int. J. Rob. Res., 35(7), pp. 823–839.
Kierzenka, J. , and Shampine, L. F. , 2001, “ A BVP Solver Based on Residual Control and the Matlab PSE,” ACM Trans. Math. Software (TOMS), 27(3), pp. 299–316.
Kebria, P. M. , Abdi, H. , and Nahavandi, S. , 2016, “ Development and Evaluation of a Symbolic Modelling Tool for Serial Manipulators With Any Number of Degrees of Freedom,” IEEE International Conference on Systems, Man, and Cybernetics (SMC), Budapest, Hungary, Oct. 9–12, p. 004223.
Herman, P. , 2009, “ Dynamical Couplings Analysis of Rigid Manipulators,” Meccanica, 44(1), pp. 61–70.

Figures

Fig. 1

Fig. 2

Material handling (left) and arc gluing (right)

Fig. 3

Heaviside-like functions defined for abstraction of tasks

Fig. 4

A schematic of the considered PUMA-like robot

Fig. 5

Optimal trajectories for the cases of: (a) normal (unbalanced), (b) static, and (c) optimal

Fig. 6

Angular velocity of the joints for the (a) normal, (b) static, and (c) optimal cases

Fig. 7

Applied torques onto the joints for the (a) normal, (b) static, and (c) optimal cases

Fig. 8

Comparison of cost values for the normal, static, and optimal cases

Tables

Table 1 Details of common repetitive tasks
Table 2 Denavi–Hartenberg parameters for the considered PUMA-like robot
Table 3 Parameters of the PUMA-like robot [17,21]
Table 4 BCs of PTP motion
Table 5 Results of PTP motion (overall time = 0.9 s)

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections