Professional Engineers (CCPE)/Manulife Financial and the Provincial Government of Ontario (OGS). References. [1] K. Youcef-Toumi. Modeling, design and ...
Task-Based Optimal Manipulator/Controller Design using Evolutionary Algorithms T. Ravichandran, G. R. Heppler and D. W. L. Wang University of Waterloo, Ontario, Canada
Abstract A design methodology, based on evolutionary algorithms, for simultaneously optimizing the topology and the parameters of a serial robot manipulator along with the required feedback control laws for performing specified tasks is presented. Dynamic performance measures, for the closed-loop system, are optimized by considering a nonlinear PD controller for an end-effector tracking problem. Results of the simultaneous design optimization involving the topology of a planar manipulator, joint motors, kinematic, inertial and control parameters as design variables illustrate the efficacy of the methodology.
Introduction Optimal design of manipulators intended for space applications is required for improving their dynamic performance measures such as energy consumption, size, weight, tracking accuracy and motion time. These performance measures can be enhanced by improving their mechanical structures and/or by using more effective controllers. Traditionally, it has been a common practice to first design the mechanical structure of a robotic system with its drive elements, and then design the controller along with the necessary trajectory planning algorithms. In recent years, design methodologies have been explored to create high performance robotic systems by simultaneously designing the mechanical structure, the actuators, the sensors, and the controller. The objective is to achieve a high performance system via a cohesive and comprehensive design approach that generates an optimal solution. Such a design approach can be truly considered to be a mechatronics design methodology [1]. A number of studies have been reported for the design of robotic manipulators with fixed topology using the simultaneous plant/controller design approach for improving dynamic performance of the closed-loop system [2, 3, 4]. Here, plant refers to the uncontrolled dynamic system and closed-loop system refers to the combination of plant and controller connected with a feedback loop. In most of these studies, the simultaneous plant/controller design problem, formulated as a parameter optimization problem using linear models and/or simple linear control strategies, is solved using numerical optimization techniques. Here, a design methodology based on evolutionary algorithms is described for simultaneously optimizing the topology and the parameters of a serial robot manipulator along with the required feedback control laws for performing specified tasks. Using this optimization based design approach,
a n-link planar manipulator and a nonlinear gain PD controller are designed simultaneously for executing end-effector trajectory tracking motions. Dynamic performance measures are optimized by searching for optimal topology, joint motors and link lengths of the manipulator along with feedback controller parameters. Results of the simultaneous design optimization are presented to illustrate the efficacy of the proposed methodology.
Simultaneous plant/controller design optimization A general methodology for simultaneous plant/controller design optimization is briefly described for closed-loop, nonlinear, mechatronic systems, like robotic manipulators. This design approach is used because the plant and controller should be designed simultaneously to achieve the best possible performance from a closed-loop system with constraints. The simultaneous design optimization problem, considers a closedloop dynamic system, of the form x˙ (t) = fs (x(t), r(t), s p , b p , sc ,t) y(t) = fo (x(t), r(t), s p , b p , sc ,t)
(1)
where t denotes time; x(t), y(t) and r(t) are the state, output and reference vectors of the closed-loop system respectively. The vectors s p and sc are the continuous parameters of the plant and the controller, respectively, the vector b p denotes the discrete parameters of the plant, and f(.) = [fs (.), fo (.)]T is, in general, nonlinear and known. Typically, the design optimization of a plant and/or its controller involves multiple and conflicting objectives which should be considered simultaneously. In this paper the approach adopted for handling multiple objectives is to form a weighted sum of the objectives into a single functional. By adopting this scheme, the simultaneous plant/controller design optimization can be formulated using functional objectives and constraints in the form min
[s p ,b p ,sc ]T ∈Ω pc
φ pc (s p , b p , sc ,t f ) =
nO
min
∑ wi φi (s p , b p , sc ,t f )
[s p ,b p ,sc ]T ∈Ω pc i=1
(2)
subject to the inequality and equality constraints: ψIj (s p , b p , sc ,t f ) ≤ 0
j = 1, . . . , nI
ψEk (s p , b p , sc ,t f ) = 0
k = 1, . . . , nE
(3)
where φi , ψIj and ψEk are design objectives, inequality constraints, and equality constraints, respectively, and t f is the evaluation time interval for the functional objectives and constraints. The feasible solution set denoted as Ω pc consists of all those solutions which satisfy all the design constraints. The non-negative numbers wi are
weights which assign the relative importance among the design objectives. The parameters nO , nI and nE represent the number of design objectives, inequality constraints, and equality constraints, respectively. The simultaneous plant/controller design optimization is a complex nonlinear optimization problem due to the non-convex, non-differentiable, multiple, dynamic objectives and constraints, and the presence of continuous and discrete design variables. Evolutionary algorithms are particularly well suited for this kind of optimization problem. Evolutionary algorithms can be considered as a broad class of stochastic optimization techniques motivated by the process of natural evolution found in biological organisms. An evolutionary algorithm maintains a population of candidate solutions for the problem at hand, and makes it evolve by iteratively applying a set of stochastic operators, known as mutation, recombination and selection [5]. In this paper, to effectively solve the simultaneous design optimization problem, we propose a hybrid method by combining two members from the family of evolutionary algorithms, namely GA and (µ + λ)-ES methods [5].
Task-specific optimal manipulator/controller design using evolutionary algorithms In this section we describe the design optimization methodology for simultaneously optimizing a manipulator and a feedback controller using evolutionary algorithms for performing specified tasks. The manipulator topology and its parameters along with required feedback controller parameters are considered as design variables for the design optimization problem. The design objectives and constraints are obtained from the given task specifications to be achieved by the optimal design of the closed-loop system. To illustrate the methodology, and for the purpose of example, we consider the design optimization of a n-link planar manipulator intended for use in space, and a nonlinear gain PD controller for accurately tracking a given end-effector trajectory while minimizing the energy consumption and subject to actuator constraints. The manipulator consists of n revolute joints actuated by n direct drive motors to rotate the links in a plane and carries a payload mass ml which is located at the distal end of the last link as shown in Fig. 1(a). The mass of each link is assumed to be distributed uniformly along the link length. For link i, li denotes the link length and mai denotes the actuator mass. In the absence of gravity, friction and other disturbances, and assuming the manipulator is attached to a very large spacecraft, the dynamic equations of the n-link rigid manipulator can be written as [6] ˙ q˙ = u M(q)q¨ + C(q, q)
(4)
where u ∈ Rn is the vector of input torques and, q ∈ Rn , q˙ ∈ Rn and q¨ ∈ Rn are vectors of joint angular positions, velocities and accelerations, respectively. M(q) ˙ q˙ is a vector representing the Coriolis and centrifugal is the inertia matrix, C(q, q) forces.
According to given task specifications, the end-effector of the manipulator is required to track the following trajectory in the Cartesian plane: xd (t) = 2.0 + 0.5 cos(θ(t)) yd (t) = 2.0 + 0.2 sin(2θ(t) + π/6)
(5)
θ(t) = 2πt/t f − 0.5 sin(4πt/t f ) where t f is the maximum travel time of the manipulator. Since the topology of the serial manipulator is considered as part of the design variables (i.e. n is not known in advance), and allowing redundancy of the manipulator, the joint angular velocities corresponding to the end-effector trajectory (5) can be given by q˙ d = J(q)† [x˙d y˙d ]T
(6)
where J(q)† is the Moore-Penrose inverse of the manipulator Jacobian J(q) and defined according to J(q)† = J(q)T (J(q)J(q)T )−1 (7) provided det(J(q)J(q)T )−1 6= 0. Having determined q˙ d (t) and given the initial angular positions qd (0), one can compute the joint angular positions qd (t) and joint angular accelerations q¨ d (t) using any reliable numerical integration and differentiation techniques, respectively, for the time interval t ∈ [0,t f ]. The objective of the manipulator design optimization is to minimize the energy consumption of the manipulator while performing end-effector tracking of the trajectory in (5). Toward this goal, the topology of the manipulator (n links connected serially) and the required joint motors (motor mass mai and maximum torque umax i ) are chosen as discrete design variables b p , and the links lengths and the initial joint angles are selected as continuous design variables s p to minimize the energy consumption given by n
φen (s p , b p , sc ,t f ) = ∑ wei i=1
Z tf 0
u2i dt
(8)
subject to the following kinematic and actuator constraints: limin ≤ li ≤ limax
and
|ui | ≤ umax i
for i = 1, . . . , n
(9)
while the end-effector follows the trajectory (5). In this example the non-negative weights wei are chosen to be 1 to give equal importance for all the links. In the manipulator design optimization, the feedback controller parameters sc are fixed to be sc = 0. The problem of trajectory tracking for robotic manipulators whose dynamics are given by (4) can be accomplished using the so-called PD+ controller [7] defined as ˙ q˙ d u = K p q˜ + Kd q˙˜ + M(q)q¨ d + C(q, q)
(10)
where qd is the vector of desired joint positions, q˜ = qd − q is the vector of tracking errors, K p and Kd are the proportional and derivative gain matrices. It has been shown
that the robot dynamic model (4) together with control law (10) is globally asymptotically stable provided that K p and Kd are symmetric, positive definite matrices, and the feed-forward term in (10) contains complete manipulator dynamics [7]. Although the stability of closed-loop systems when using the control law in (10) is assured, the system performance is dictated by the choice of controller gains K p and Kd , which are assumed to be constant. To achieve high performance or to satisfy real constraints of actual manipulators, it may be necessary to have variable controller gains. In [8], a class of nonlinear gain PD controllers was proposed for trajectory tracking control serial n-link rigid robotic manipulators whose dynamics are given by equation (4). The nonlinear gain PD controllers of this class are defined as ˙˜ q˙˜ + M(q)q¨ d + C(q, q) ˜ q˜ + Kd (q) ˙ q˙ d u = K p (q)
(11)
˙˜ are ˜ and the derivative gain matrix Kd (q) where the proportional gain matrix K p (q) symmetric and positive definite, and have the following structure: h i k k ˜ = diag a +|p1q˜ | . . . a +|pnq˜ | K p (q) n n 1
˙˜ = diag Kd (q)
h
1
kd1 b1 +|q˙˜1 |
...
kdn bn +|q˙˜n |
i
where k pi , kdi , ai , and bi are positive constants. It has been shown that the closedloop system of the manipulator model (4) with the controller in (11) is globally exponentially stable for the above selection of nonlinear gain matrices [9]. When the feed-forward term of the controller in (11) does not contain complete manipulator dynamics, the stability and robustness analysis has been demonstrated by Ravichandran et al. [10] by establishing sufficient conditions for the exponential convergence and uniform ultimate boundedness [6] of the tracking errors. These sufficient conditions can be used to derive the lower bound smin c for the controller parameters sc (see (13)). The objective of the controller design optimization is to minimize the energy consumption and the end-effector tracking errors for the manipulator while following the Y Segment n (SEn )
r(t)
Level−1 Genes (topology) qn mn , ln , In
SE1
SEN
mn−1 , ln−1 , In−1 qn−1
Segment 1 (SE1 )
m2 , l2 , I2
m1 , l1 , I1
s11
s1n1
sN 1
sN nN
σ11
1 σn1
σ1N
N σnN
Segment n−1 (SEn−1 )
q2
Segment 2 (SE2 ) X
q1
Level−0 Genes (parameters)
(a)
(b)
Figure 1: Planar n-link manipulator for specified-task
trajectory in (5). The feedback controller parameters k pi , kdi , ai , and bi are chosen as the design variables sc to minimize the energy consumption in (8) and the sum of end-effector tracking errors defined as φac (s p , b p , sc ,t f ) =
Z tf
(x˜2 + y˜2 )dt
(12)
0
where x˜ = xd −x and y˜ = yd −y are the deviations between the actual (x, y) and desired (xd , yd ) end-effector positions, and subject to max smin c ≤ sc ≤ sc
(13)
along with the actuator constraints as given in (9). In the controller optimization, the manipulator parameters s p and b p are fixed at the optimal values found during the manipulator design optimization. Because the simultaneous design optimization of the manipulator and the feedback controller involves both discrete (manipulator topology and actuators) and continuous (link lengths, initial joint angles and controller parameters) design variables we will use a hybrid evolutionary algorithm (GAES) which is formulated by combining the concepts from two well known members of evolutionary algorithms family, namely genetic algorithms (GA) and evolution strategies (ES) [5] to effectively perform the design optimization. Like any other evolutionary algorithm, the basic components of GAES are representation of candidate solutions, initial population, genetic operators, fitness evaluation and termination condition. Each candidate solution representing the topology and parameters of the manipulator and the feedback controller parameters are encoded into a multi-level chromosome (MLC) as shown in Fig. 1(b) so that it can be manipulated by the GAES algorithm. The topology of the manipulator is constituted by a number of basic units called segments (SEi ) where a segment is formed by combining a joint, a link, and the associated parameters (e.g. motor parameters, controller parameters, the link length and the initial joint angle) together (see Fig. 1(a)). The topology of the manipulator is encoded into level-1 genes of the MLC using a binary string of length N = 10 where SEi = 1 or 0 indicates the presence or absence of the i-th link thus allowing the representation of different topologies with 2 ≤ n ≤ N links. Using the representation schemes adopted in canonical ES methods [5], the parameters si and their self-adapted mutation steps σi are encoded into level-0 genes of the MLC as block-wise arrays of real values (see Fig. 1(b)). An initial population of µ parents is generated randomly and using GA-like crossover and mutation operations on the level-1 genes of the parent population, an intermediate µ number of parents are generated. Using ES-like recombination and mutation operations on the level-0 genes of the intermediate parent population, an offspring population of λ (> 5µ) individuals are generated. The fitness evaluation of the offspring population is performed using the objective functions in (8) and (12) subject to the constraints in (9) and (13). Having obtained the fitness values for the offspring population, the best µ individuals are selected as a new parent population and the generation cycle is repeated until some maximum number of generations are performed.
Manipulator/controller design optimization results Sample results for the optimization procedure and example case described above are given for a payload of 0.5 kg. The desired motion of the end-effector is described by (5) with t f = 5 s. The optimization procedure GAES searches for the best topology of the n-link planar manipulator (2 ≤ n ≤ 10) and the required joint motors from the following discrete set: (mai , umax i ) ∈ {(0.25, 5.0), (0.5, 10.0), (1.0, 20.0), (1.5, 30.0), (2.5, 50.0)} along with other manipulator and controller parameters which take continuous values from their specified ranges. The simultaneous design optimization objective is to minimize the energy consumption and the end-effector tracking error by choosing simultaneously the manipulator topology, the joint motors, the manipulator parameters s p , consisting of link lengths and the initial joint angles, and the feedback controller parameters sc , subject to design constraints. This optimization problem can be stated using the weighted-sum formulation in (2) with φ pc defined as φ pc (s p , b p , sc ,t f ) = wen φen (s p , b p , sc ,t f ) + wac φac (s p , b p , sc ,t f )
(14)
and the design constraints given in (9) and (13). The non-negative weights wen and wac indicate the relative importance of minimizing the energy consumption φen as compared to the tracking errors φac . l1 (m) 1.176 1.303
seq sim
seq sim
seq sim
k p1 100.0 99.92 ma2 (kg) 0.5 0.5
l2 (m) 0.634 1.081
k p2 100.0 99.99
umax 2 (Nm) 10.0 10.0
l3 (m) 0.658 1.089
k p3 100.0 86.35
ma3 (kg) 0.5 0.25
l4 (m) 0.971 -
k p4 100.0 -
umax 3 (Nm) 10.0 5.0
ma4 (kg) 0.25 -
qo1 (rad) 0.505 1.004
kd1 52.22 99.95 umax 4 (Nm) 5.0 -
kd2 2.0 2.003
qo2 (rad) 0.870 kd3 2.0 5.003
kd4 2.875 -
φen
φac
φ pc
0.482 0.058
262.0 264.2
0.492 0.176
Table 1: Comparison in design variables and performance To demonstrate the effectiveness of the simultaneous design methodology using the proposed optimization procedure, the following design optimization studies were performed: Sequential design optimization (seq-opt): First, the manipulator design optimization was carried out with the weights chosen as wen = 1 and wac = 0. The design variables consist of discrete variables b p representing the manipulator topology and joint motors as described above, and continuous variables s p representing link lengths li for i = 1, . . . , N and initial joint angles qoi for i = 1, . . . , N − 2 (qoN−1
and qoN are determined using inverse kinematics). The feedback controller parameters sc = 0 were fixed and the control law in (10) was used with the feed-forward term consisting of complete manipulator dynamics. The constraints in (9) were used = 2.0 rad. Then the = 0.0 rad and qmax with limin = 0.2 m, limax = 2.0 m, qmin i i controller optimization was performed with the weights chosen as wen = 0.25 and wac = 0.75. Now, the design variables consist of only the feedback controller gains sc = [k p1 , . . . , k p10 , kd1 , . . . , kd10 ]T with ai and bi taken as 0.05 and 0.1, respectively, while s p and b p are fixed at the optimal values found in the manipulator optimization. A nominal payload of 0.25 kg was considered for the feed-forward part of the T control law in (10). The constraints in (13) were used with smin c = [2.0, . . . , 2.0] and T max sc = [100, . . . , 100] . Simultaneous design optimization (sim-opt): The manipulator and the controller design optimization cases from the sequential approach were considered together by using all the design variables and the constraints simultaneously along with the same weights as used in the controller optimization before.
Sequentialy optimized manipulator topology and tracking performance
Simultaneously optimized manipulator topology and tracking performance
2.5
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2
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link 4 link 3
Y [m]
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(a)
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(b)
Comparison of end−effector x−y position errors
x 10
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Motor torques for simultaneously optimized manipulator and controller 20
xerr−seq yerr−seq xerr−sim yerr−sim
3
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End−effector x−y position error [m]
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(c)
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Joint 1 Joint 2 Joint 3 0
0.5
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(d)
Figure 2: Optimal manipulator and controller performance for specified-task
5
The GAES method was used to solve the above design optimization cases with µ = 15 and λ = 100. The results obtained from these studies are given in Table 1. The optimized manipulator topology and the end-effector tracking performance obtained from the sequential design optimization are depicted in Fig. 2(a). Fig. 2(b) shows the same for the simultaneous design optimization. The end-effector tracking errors in x and y positions are compared in Fig. 2(c) for the sequential and simultaneous design cases. Fig. 2(d) shows the joint motor torques for the optimal design obtained from the simultaneous design optimization. It can be noted from these results that the simultaneous design optimization yields a different manipulator topology and design parameter values than those obtained by sequential design optimization (see Table 1, Fig. 1(a) and Fig. 2(b)). As indicated by the objective function values obtained in these design cases, the performance of the closed-loop system for specified tasks can improved significantly by simultaneously optimizing the topology and the parameters of the manipulator along with the controller parameters.
Summary This paper has presented an effective design optimization methodology based on evolutionary algorithms for simultaneously optimizing the topology and the parameters of a serial robot manipulator along with the required feedback control laws. As an example, simultaneous design optimization of a planar manipulator and a nonlinear gain PD controller for a trajectory tracking task has been investigated. It has been demonstrated by the simulation results that the simultaneous design optimization methodology can lead to a more desirable design of the manipulator and the controller for specified tasks.
Acknowledgment This work has been supported partially by scholarships from the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canadian Council of Professional Engineers (CCPE)/Manulife Financial and the Provincial Government of Ontario (OGS).
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[3] Rastegar, J. S., Liu, L., and Yin, D. Task-specific optimal simultaneous kinematic, dynamic, and control design of high-performance robotic systems. IEEE/ASME Transactions on Mechatronics, 4(4):191–203, 1999. [4] Li, Q., Zhang,W. J., and Chen, L. Design for control - a concurrent engineering approach for mechatronic systems design. IEEE/ASME Transactions on Mechatronics, 6(2):161–169, 2001. [5] Back, T. Evolutionary Algorithms in Theory and Practice. Oxford University Press, New York, 1996. [6] Spong, M. W. and Vidyasagar, M. Robot Dynamics and Control. John Wiley, New York, 1989. [7] Paden, B. and Panja, R. Globally asymptotically stable PD+ controllers for robotic manipulators. International Journal of Control, 47(6):1697–1712, 1988. [8] Kelly, R. and Carelli, R. A class of nonlinear PD-type controllers for robotic manipulators. Journal of Robotic Systems, 13(12):793–802, 1996. [9] Ravichandran, T., Heppler, G. R., and Wang, D. W. L. Stability analysis of a class of nonlinear controllers. In Symposium on Advances in Robot Dynamics and Control, CD Proceedings of the ASME International Mechanical Engineering Congress and R&D Expo, volume 2, Washington, D.C., November 15-21 2003, Paper Number: IMECE2003-42863. [10] Ravichandran, T., Wang, D. W. L., and Heppler, G. R. Stability and robustness of a class of nonlinear controllers for robot manipulators. In Proceedings of the American Control Conference, Boston, MA, June 30 - July 2 2004 (to appear).
