Abstract
Purpose
This paper aims to address a new iterative tuning method of PID control for robot manipulators.
Design/methodology/approach
This tuning method uses several properties of the robot control, such as any PD control can stabilize a robot in regulation case, the closed-loop system of PID control can be approximated by a linear system, the control torque to the robot manipulator is linearly independent of the robot dynamic.
Findings
Compared with the other PID tuning methods, this novel method is simple, systematic, and stable. The transient properties of this PID control are better than the other normal PID controllers.
Originality/value
In this paper, a new systematic tuning method for PID control is proposed. The paper applies this method on an upper limb exoskeleton, and real experiment results give validation of our PID tuning method.
Keywords
Citation
Yu, W., Li, X. and Carmona, R. (2013), "A novel PID tuning method for robot control", Industrial Robot, Vol. 40 No. 6, pp. 574-582. https://doi.org/10.1108/IR-09-2012-406
Publisher
:Emerald Group Publishing Limited
Copyright © 2013, Emerald Group Publishing Limited
1 Introduction
Proportional-integral-derivative (PID) control is widely used in industrial robot manipulators. In the absence of the robot knowledge PID control may be the best controller, because it is model-free, and its parameters can be adjusted easily and separately (Ang et al., 2005; Astrom and Hagglund, 2004). The great advantages of PID control over the others are they are simple and have clear physical meanings. However, the PID gains must be tuned to guarantee good performances such as rise-time, overshoot, settling time, and steady-state error.
Since PID controller is in linear form, the main study on PID tuning focused on linear systems (O'Dwyer, 2006). The tuning process of PID gains can be classified into five categories:
Model-based analytical tuning. According to analytical relations between the model and the control objective, PID gains are calculated from some algebraic equations (Chang and Jung, 2009; Chien and Fruehauf, 2002; Juang et al., 2008).
Heuristic methods. These methods combine several techniques, such as practical experience (Ziegler and Nichols, 1942; Cohen and Coon, 1953), manual tuning (Astrom and Hagglund, 2004), and artificial intelligence (Sun and Er, 2004; Lewis et al., 1995; Juang et al., 2008).
Frequency domain methods. Frequency characteristics are easy to be obtained for linear systems. When the controlled process is a linear system, PID controller can be tuned in frequency domain (Shinskey, 1996).
Optimization methods. PID control is transformed into a special optimal control form. PID tuning becomes an off-line numerical optimization problem (Kristiansson and Lennartsson, 2002).
Adaptive methods. Base on adaptive control and parameter on-line identification, PID gains are tuned as an automated tuning process (Wang et al., 2001).
PID tuning in robot control can be grouped as:
Intelligent methods. The intelligent techniques, such as fuzzy logic (Sun and Er, 2004), neural networks (Lewis et al., 1995) and genetic method (Juang et al., 2008), are used to tune PID gains. The final controllers are no longer industrial linear PID.
Impedance control. The inverse dynamic is applied to transfer the robot into a linear system. Then mechanical impedance idea was applied to tune PID gains (Hogan, 1985). PID gains could be adjusted by discrete-time approximation of the inverse dynamic (Chang and Jung, 2009).
Lyapunov approach. Lyapunov approach was used to adjust PID controller such that the robot controller could follow a linearization control (Chang et al., 1986).
There are the following difficulties to design a systematic tuning method for robot PID control:
The control torque of each joint affects the other one, and these influences are strong nonlinear.
There are too many gains to be tuned simultaneously and heuristically for robot via Ziegler-Nichols (Ziegler and Nichols, 1942) or Cohen-Coon (Cohen and Coon, 1953) methods. For example, a six degree-of-freedom robot manipulator has 18 gains to be tuned. When one gain is tuned, it requires tuning the other 17 gains in turn because of coupling dynamic.
Some nonlinear methods, such as stability analysis, can obtain the upper and lower bounds of PID gains. However, the desired performances are not guaranteed.
In regulation case, a robot can be stabilized by any PD controller providing that the PD gains are positive.
The closed-loop behaviors of robot PID control are similar with linear systems.
The control torque of each joint is independent of the robot dynamic.
2 Closed-loop tuning for robot manipulators
The dynamics of a serial n-link rigid robot manipulator can be written as Spong and Vidyasagar (1989): Equation 1 where q∈Rn denotes the links positions, q˙∈Rn denotes the links velocity, M(q)∈Rn×n is the inertia matrix, C(q,q˙)∈Rn×n is the centripetal and Coriolis matrix, g(q)∈Rn is the gravity vector, f∈R n is the frictional terms (Coulomb friction), and τ∈Rn is the input control vector. The model (1) can be written as: Equation 2 If the model is completely known, a nonlinear feedback linearization controller is Spong and Vidyasagar (1989): Equation 3 The closed-loop system becomes a linear system: Equation 4 If some parameters of the robot are unknown, the robot model can be written in the parameter-in-linear form Craig et al. (1986): Equation 5 where the parameter is unknown, the adaptive control can be applied as: Equation 6 The above two types of controllers are model-based. They depend on the robot model (1). When the model is unknown, the model-free PID control should be used.
Since the robot dynamic is not stable in open loop, it is impossible to send a step command to all joints of the robot to tune PID gains. In this paper we use the following closed-loop tuning method.
2.1 Stable PD control
The classical industrial PD law is: Equation 7 where K p and K d are positive definite, symmetric and constant matrices, which correspond to proportional and derivative coefficients, q d∈Rn is the desired joint position, q˙ d∈Rn is the desired joint velocity. In the paper we discuss regulation case. The desired position is constant, i.e. q˙ d=0.
In the control for robot manipulators, the desired joint positions are generated by the path planning or trajectory planning, and are sent to the joint motors (Kelly et al., 2005). The tracking control is divided into several regulations by the path planning. The purpose of a tracking control is to make the joint motors to follow the desired positions.
Now we analyze stability property of the PD control. We use a Lyapunov function candidate as: Equation 8 For any robot, the inertia matrix M(q) and the Coriolis matrix C(q,q˙) satisfy the following condition (Spong and Vidyasagar, 1989): Equation 9 For any two vectors q˙ and (g+f), they satisfy (Kelly et al., 2005): Equation 10 where K 1 is any positive definite matrix. Using equations (1) and (2), the derivative of equation (3) is: Equation 11 where (g+f)T K 1−1(g+f)≤d¯, d¯ can be regarded as upper bound of (g+f). If we choose K d >K 1, the regulation error q˜ is bounded (stable), and ‖q˙‖(K d −K 1) converges to d¯.
So the position regulation error of the PD control law (equation (2)) is bounded in a ball with radius d¯. However, the stability is not enough for robot control. The steady-state error caused by gravity and friction may be large.
We use the stability property of PD control (equation (2)) to stabilize the open loop unstable robot (equation (1)). From the above proof we know when K d −K 1>0 (K 1>0), the following closed-loop system is stable: Equation 12 The manipulator dynamic contains the gravitational torque vector g(q). For heavy manipulators, gravity compensation is a popular method to modify the PD control (equation (2)). The new PD control is: Equation 13 where g(q)=g^(q)+g˜(q), g^(q) and g˜(q) are the gravity estimation and estimation error. In this case (equation (4)) become: Equation 14 where d¯ 1 is the upper bound of (g˜+f), (g˜+f)T K 1−1(g˜+f)≤d¯ 1. Normally d¯ 1≪d¯, because g˜ is the estimation error of the gravity. The stable closed-loop system (5) becomes: Equation 15
2.2 PID tuning in closed-loop
The PD control (equation (6)) cannot guarantee zero of the steady-state error. The integrator is the most effective tool to eliminate steady-state error. PD control (equation (2)) becomes PID control: Equation 16 where K p , K i and K d are proportional, integral and derivative gains of the PID controller, respectively. The integrator gain K i should be increased when the steady-state error is large. This causes large overshoot, long settling time, and less robust.
Now we show how to tune PID2 independently for the stable closed-loop system (7). When we apply a PID control to the closed-loop system (5), it is: Equation 17 When we apply a gravity compensation to the closed-loop system (5), it is: Equation 18 When we apply the PID control and the gravity compensation to the closed-loop system (7), it is: Equation 19 The total control torque to the robot is: Equation 20 From equations (9) to (12), we see that the control torque to the robot manipulator is linearly independent of the robot dynamic (equation (1)). In general case, if we tune PID controllers m times, they can be expressed as: Equation 21 where: Equation 22 PD1 is a special PID with K i =0. This property allow us to start a PID control with small gains, such that the closed-loop system is stable. Then any other tuning rule can be applied to obtain new PID gains independently. The final PID gains are the summarization of all these controllers (gains).
2.3 Linearization of the closed-loop system
Although PID2 can help to reduce steady state error of PD1, the above sub-section does not provide a systematic tuning method for the PID gains. Inspired by the well known PID tuning methods for linear systems, we will derive a similar method in this section.
The robot dynamics are strong nonlinear, and the behaviors of the closed-loop system with PD/PID control are similar with the transient responses of a linear system. On the other hand, after PID control, each joint of the robot can be characterized as a single input-single output (SISO) system.
Several methods can be used to linearize the robot models. If the velocity and gravity are neglected, the terms C(q,q˙)q˙ and g(q) in the nonlinear dynamics (equation (1)) are zero. The resulting system is a linear model (Goldenberg and Bazerghi, 1986): Equation 23 Obviously, it is an over-simplified model.
Since the velocity dependent term C(q,q˙)q˙ representing Coriolis-centrifugal forces, it is negligible for small joint velocities. A rate linearization scheme can be used as Golla et al. (1981): Equation 24 where: Equation 25 q 0 is operating point. But many experiments show that even at low speeds, C(q,q˙) is not zero (Swarup and Gopal, 1993).
The velocity and gravity are main control issues of robots, and they are dominant components of the dynamics. When the robot model is completely known, Taylor series expansion can be applied (Li, 1989). At the operating point q 0, the nonlinear model (1) can be approximated by: Equation 26 where: Equation 27 In this paper, we use identification-based linearization method. For each joint, typical linear model with PD/PID control is a first-order system with transportation delay as: Equation 28 The response of the model (17) is characterized by three parameters, the plant gain K m , the delay time t m , and the time constant T m . These are found by drawing a tangent to the step response at its point of inflection and noting its intersections with the time axis and the steady state value (Figure 2).
Sometimes the first-order model (17) cannot describe the complete nonlinear dynamic of robot. Another linear model of robot is Taylor series model as in equation (16). The model can be written in frequency domain: Equation 29 or: Equation 30 The responses of this second-order model are similar with mechanical motions. If there exists a large overshoot, a negative zero is added in equation (18): Equation 31
2.4 PD/PID tuning
Because the robot can be approximated by a linear system, some tuning rules for linear systems can be applied for the closed-loop system tuning. The normal input signals for PID tuning are step and repeat inputs.
First we derive PD tuning rules. When each joint can be approximated by a first-order system, the approximated model is: Equation 32 Here K m , T m and t m are obtained from Figure 2.
The linear PID law in time domain (equation (8)) can be transformed into frequency domain: Equation 33 Some popular PID tuning methods, such as Ziegler-Nichols (Ziegler and Nichols, 1942) and Cohen-Coon (Cohen and Coon, 1953) methods, are based on this transfer function. By several experiments, we propose our PD gains tuning method as in Table I.
If each joint is approximated by a second-order system: Equation 34 the PD gains are tuned as in Table II.
Here Model 1 is from Huang et al. (2005), and Model 2 is from Chien and Fruehauf (2002).
If PD control cannot provide good performances, PID control should be used. The PID gains for the first-order model are shown in Table III.
Here K c =K p is proportional gain, T i =K c /K i is integral time constant and T d =K d /K c is derivative time constant.
The PID gains for the second-order model is decided by Table IV.
2.5 Refine PID gains
From equation (13) we see the PID gains are linear independent, we can modify them directly. In this way, a new PID controller, PID3 is included. We use Table V to refine PID3.
After the refined process, the robot control is: Equation 35 In order to decrease the steady-state error, we should increase K i . Decreasing K d gives less settling time, and decreasing K p provides less overshoot. However, the above tuning process does not guarantee stability of the closed-loop system. In the next sub-section, we will give the bounds of the PID gains to ensure the stability of the PID control.
2.6 Stability conditions for PID gains
It is not easy to obtain stability conditions for the linear PID control for the robots, because they do not include any nonlinear component of the robot dynamic (Kelly et al., 2005). In order to assure asymptotic stability of PID control, the simplest approach is to modify the linear PID into nonlinear one.
A few research works on the most popular industrial controller, linear PID. In Pocco (1996) the robot dynamic was rewritten in decoupled linear system and bounded nonlinear system; this linear PID control could not guarantee asymptotic stability. Sufficient conditions of the linear PID in Kelly et al. (2005) were given via Lyapunov analysis. However, these conditions are not explicit, the PID gains could not be decided with these conditions directly, a complex tuning procedure was needed (Kelly, 1995).
Without considering PD1 PID2 and PID3, the PID control law (equation (8)) can be expressed via the following equations: Equation 36 The closed-loop system of the robot (equation (5)) is: Equation 37 The equilibrium is [ξ,q˜,q˜˙]=[ξ*,0,0]. Since at equilibrium point q=q d, the equilibrium is [g(q d),0,0]. In order to move the equilibrium to origin, we define ξ˜=ξ−g(q d). Closed-loop equation becomes ξ˜˙=K i q˜: Equation 38 In Yu and Rosen (2010) we have proven that the robot dynamic controlled by the linear PID controller (equation (21)), the closed loop system (22) is semi globally asymptotically stable at the equilibrium x=[ξ−g(q d),q˜,q˜˙]T=0, provided that control gains satisfy: Equation 39 where β=√(λ m (M)λ m (K p ))/3, k g satisfies ‖g(q 1)−g(q 2)‖≤k g ‖q 1−q 2‖, λ M (M) is the maximum eigenvalue of M, λ m (K p ) is the minimum eigenvalue of K p . The maximum eigenvalue of M can be estimated without knowing M (Kelly et al., 2005).
The three gain matrices of the linear PID control (equation (21)) can be chosen directly from the conditions (23). From equation (13) we know the PID control with gravity compensation (equation (20)) is: Equation 40 Now we apply the condition (23) to PID f . If the gains of PID f are not in the bound of equation (23), we add a new PID controller, PID4, such that the gains of PID f + PID4 are in the bounds of equation (23). The final control torque to the robot is: Equation 41 It is also explained in Figure 1.
3 Application to an exoskeleton
A research group in UCSC has successfully constructed a 7-DOF exoskeleton robot (Figure 3). In this paper, we apply our novel PID control in this exoskeleton. The computer control platform of the UCSC 7-DOF exoskeleton robot is a PC104 with an Intel [email protected] GHz processor and 512 Mb RAM. The motors for the first four joints are mounted in the base such that large mass of the motors can be removed. Torque transmission from the motors to the joints is achieved using a cable system. The other three small motors are mounted in link five.
Fortunately, this upper limb exoskeleton is fixed on the human arm, the behavior of the exoskeleton is the same as the human arm (Figure 4). It is composed of a 3-DOF shoulder (J1-J3), a 1-DOF elbow (J4) and a 3-DO wrist (J5-J7). J1-J3 are responsible for shoulder flexion-extension, adduction and internal-external rotation, J4 creates elbow flexion-extension, J5-J7 are responsible for wrist flexion-extension, pronation-supination and radial-ulnar deviation. We regard J1, J2 and J3 in Figure 4 as three spherical joints of the human shoulder (Figure 4). Also J5, J6 and J7 in Figure 4 are considered as three spherical joints of the human wrist.
The real-time control program operated in Windows XP with Matlab 7.1, Windows Real-Time Target and C++. All of the controllers employed a sampling frequency of 1 KHZ. The properties of the exoskeleton with respect to base frame are shown in Table VI.
We first use the following PD1 to stabilize the robot: Equation 42 The joint velocities are estimated by the standard filters: Equation 43 Here the main weight of the exoskeleton is in the first four joints. The potential energy is: Equation 44 The gravity compensation in equation (22) is calculated by: Equation 45 We use the step responses of linear systems to approximate the closed-loop responses of the robot via PD1. The step responses of the closed-loop systems of the robot and the linear systems are shown Figure 5, where in the dash lines are the step responses of the following second-order linear systems: Equation 46 In order to tuning PID gains for the linear systems (25), we rewrite the PID (equation (8)) as: Equation 47 where K c =K p is proportional gain, T i =K c /K i is integral time constant and T d =K d /K c is derivative time constant. We use the following tuning rule: Equation 48 To tune the PID parameters. This rule is similar with Huang et al. (2005) and Chien and Fruehauf (2002), in their case: Equation 49 It is different with the other two famous rules, Ziegler-Nichols and Cohen-Coon methods, where: Equation 50 Because their rules are suitable for the process control, our rule is for mechanical systems. By the rule (equation (26)), the PID2 is: Equation 51 The control torque becomes u=PID1+g^(q)+PID2.
Since the trajectory control is realized by several regulations. Each regulation needs a quick response. When we do the off-line PID tuning, we give fast reference signal, and now robot is not contracted with human. The control results of Joint 1 are shown in Figure 6.
After this refine turning, PID3 is: Equation 52 The final control is: Equation 53 The stability condition (23) gives a sufficient condition for the minimal values of proportional and derivative gains and maximal values of integral gains. We find that the final control (equation (29)) satisfies the conditions (23), it is PID4.
The final control results for Joint 3 and Joint 7 are shown in Figure 7.
We use Figure 8 to show the affection of the gravity compensation. The curve “With gravity compensation” is PID f =PD 1+g^(q)+PID 2+PID 3. The curve “Without gravity compensation” is PID f =PD 1+PID 2+PID 3.
Now we use Joint 2 to explain our PID tuning process. The control results of the different PID controllers are shown in Figure 9. It can be found that each PID tuning step improve the control behavior.
4 Conclusions
This paper proposed a systematic multi-step tuning method for robot PID control. The roles of the four PID controllers are:
PD 1 does not need to be tuned. We show that any PD 1 can stabilize the robot. The role of PD 1 is to derive a stable closed-loop system.
The main tuning process is in PID 2. Because the behavior of the closed-system with PD 1 is similar with the linear system, several tuning methods are used for PID 2 tuning.
PID 3 is used to refine the control result under PD 1 and PID 2. Here we use an important property of robot control. Tuning the gains of PID 3 is the same as tuning whole PID gains.
The role of PID 4 is to adjust the gains of PD 1+PID 2+PID 3 to meet the stability criterion. Using the above property, the adjustment of PD 1+PID 2+PID 3 is the same with adding PID 4.
About the authors
Wen Yu received the BS degree from Tsinghua University, Beijing, China in 1990 and the MS and PhD degrees, both in electrical engineering, from Northeastern University, Shenyang, China, in 1992 and 1995, respectively. From 1995 to 1996, he served as a Lecturer in the Department of Automatic Control at Northeastern University, Shenyang, China. In 1996, he joined CINVESTAV-IPN, Mexico, where he is currently a Professor in the Departamento de Control Automatico. He held research positions with the Instituto Mexicano del Petroleo from 2002 to 2003. He was a Senior Visiting Research Fellow at Queen's University Belfast from 2006 to 2007, a visiting Associate Professor at University of California Santa Cruz from 2009 to 2010. He also holds a visiting professorship at Northeastern University in China from 2006 until now. Dr Wen Yu serves as an Associate Editor of Neurocomputing, and Journal of Intelligent and Fuzzy Systems. He is a member of Mexican Academy of Science. Wen Yu is the corresponding author and can be contacted at: [email protected]
Xiaoou Li received her BS and PhD degree in applied mathematics and electrical engineering from Northeastern University, China, in 1991 and 1995. From 1995 to 1997, she was a Lecturer of electrical engineering at the Department of Automatic Control of Northeastern University, China. From 1998 to 1999, she was an Associate Professor of computer science at the Centro de Instrumentos, Universidad Nacional Autónoma de México (UNAM), México. Since 2000, she has been a Professor of the Departamento de Computación, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (CINVESTAV-IPN), México. During the period from September 2006 to August 2007, she was a visiting Professor of School of Electronics, Electrical Engineering and Computer Science, the Queen's University of Belfast, UK. Her research interests include Petri net theory and application, neural networks, knowledge based system, and data mining.
Roberto Carmona received the BS degree in mechanical engineering from Orizaba Technologic Institute, Veracruz, Mexico, in 2001 and the MS degree in automatic control from CINVESTAV-IPN, México, in 2005. He is currently pursuing the PhD degree in the Department of Computing, CINVESTAV-IPN. His research interests include support vector machine, pattern classification, neural networks, fuzzy logic and clustering.
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Further Reading
Cominos, P. and Munro, N. (2002), “PID controllers: recent tuning methods and design to specifications”, IEE Proceedings – Control Theory and Applications, Vol. 149 No. 1, pp. 46-53.
Li, X. and Yu, W. (2011), “A systematic tunning method of PID controller for robot manipulators”, 9th IEEE International Conference on Control & Automation, ICCA11, Santiago, Chile, pp. 274-279.