Proceedings of the 2000 IEEE/RSJ International Conference on Intelligent Robots and Systems
Instantaneous Model Impedance Control for Robots Tomer Valency, and Miriam Zacksenhouse Sensory-Motor Integration Laboratory Faculty of Mechanical Engineering Technion - Israel Institute of Technology e-mail: valency,
[email protected]
Abstract Impedance control facilitates the performance of tasks that involve contact with the environment. Two generic strategies for implementing impedance control have been described in the literature: the dynamic based approach [4] and the position based approach [7]. The second method has been introduced to eliminate the inherent sensitivity of the first method to the accuracy of the dynamic model. However, it introduces impedance errors whenever the actual position of the robot differs from that of the model. We propose a new method for implementing impedance control, which is designed to take advantage of the error-correction capabilities of position controllers, while maintaining good impedance tracking performance. The proposed Instantaneous Model Impedance Control re-initializes the impedance model to the current position of the robot so the model does not accumulate position errors. The novelty of the method is in using the position feedback both in the outer loop, to track the desired impedance, and in the inner loop to improve robustness. The proposed method includes the dynamic-based and the position-based methods as specific cases, and can be tuned to trade between their corresponding merits. Using simulations, we demonstrate the performance of the new method in comparison with the previous methods. We consider two scenarios: stiff environment, which challenges the position based approach, and load uncertainties, which challenges the dynamic based approach.
1. Introduction Impedance control has been proposed by Hogan [4] as a fundamental strategy for manipulating robots in free and constrained motion. It regulates both the force and trajectory errors to endow the robot end-effector with compliant properties that may be inherently different from its mechanical properties. The acquired properties may be designed to enhance the capabilities of the robot in contact tasks. Thus, impedance control is especially beneficial in contact tasks [4], co-operating robots [1], and assembly tasks [9] [10]. However, despite its important benefits to robotic application this control strategy is not widely used. That is mainly due to difficulties in controlling the desired
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impedance. These difficulties may be attributed to lack of accurate dynamic models and deficiencies in the proposed compensation mechanisms. The current paper introduces a novel method, which is targeted at solving these difficulties. The most popular method for implementing impedance control was introduced by Hogan [4] in 1985. This method depends inherently on the dynamic model of the robot and is therefor termed Dynamic Based Impedance Control (DB-IC). It relies on causing the robot endeffector to accelerate in the amount required to fulfil the impedance equation, given the measured position, velocity and interaction force. The control force that is required to impose the desired acceleration depends on the dynamics of the robot. Thus, this method relies inherently on an accurate model of the robot's dynamics and is very sensitive to model inaccuracies. The alternate method that was introduced by Lawrence [7] implements the impedance control by tracking the position of the desired impedance model. The impedance model is subjected to the same interaction forces as those measured at the robot end-effector, but is otherwise independent of the actual position of the robot. The desired position vector is tracked using an internal position controller, and the method is known as "inner outer control" or Position Based Impedance Control (PBIC). This method eliminates the inherent dependency on the dynamic model of the robot. The internal position controller may use the dynamic model of the robot but, since it includes error correction mechanism, is less sensitive to model inaccuracies. However, the method introduces impedance errors whenever the actual position of the robot differs from that of the model. Such position discrepancies often result from controller errors or external, unmeasured disturbances, which are especially important in contact tasks. Using an adaptive control method [8] in the inner control loop does not eliminate that problem. Essentially, the large gain in the position controller introduces high stiffness, which may have adverse effects especially in contact tasks. Thus, using inner-loop stiff robust controller increases the impedance error. The impedance errors prevent offline analytic planning of the desired impedance along the trajectory. Thus, the applicability of the method is restricted to simple tasks in which the impedance can be plane
In the current paper we propose a new method for implementing impedance control, which is designed to take advantage of the error-correction capabilities of position controllers, like PB-IC, while maintaining good impedance tracking performance, like the DB-IC.
desired impedance. The impedance is defined by the following equation:
2. Principle of Operation
stiffness, damping and mass, respectively, x0 , x0 are the
Fint = K (x 0 − x m ) + B (x 0 − x m ) − Mx m
(1)
Where Fint is the interaction force, K , B , M are the
Instantaneous model impedance control (IM-IC) combines the dynamic-based and position-based impedance control methods. It calculates the acceleration needed to fulfill the impedance relationship as in the dynamic-based approach. Using the acceleration and the current robot position and velocity it predicts the location of an instantaneous model in the next time step. That position is then converted to the joint space and used as a reference to the inner control loop. The inner control algorithm can be any algorithm that assures stable trajectory tracking. In this work, a proportionaldifferential (P.D.) plus inverse dynamics algorithm is used for direct comparison with the other impedance control methods. This overall structure is shown in Figure 1, and while it seems similar to that of the position-based method, it is unique in the use of the current position in computing of the desired trajectory in the external loop, as further detailed below. The desired trajectory is computed by integrating the impedance model in the task space for a short prediction period ( ∆t ). However, instead of integrating from the current position of the model, the integration is reinitialized to the current position of the robot. Specifically, the initial position is computed from the measured joint positions using forward kinematics. Thus, the desired position is obtained by integrating the impedance model from the current position of the robot. The desired task-space position is transferred back to the joint space, using the robot inverse kinematics and its Jacobian, and provides a reference signal to the innerloop position controller.
3. Mathematical Formulation
xim (t ) = L(qr ) ; xim (t ) = Jqr
(2)
q r , q r are the joint position and velocity vectors of the real robot, L(⋅) represents the forward kinematics of the robot, and J its Jacobian. As in the dynamicWhere
based method, the acceleration needed to fulfil the desired impedance in Equation (1) is computed based on the measured force, position and velocity. The acceleration is integrated twice to generate a reference position for the inner control loop. To achieve an accurate impedance tracking we initialize the integration constants according to the current robot position. This feature corresponds to having an instantaneous impedance model that is sensitive to actual robot position instead of a position independent impedance model as previously suggested in the literature [7], [8], [10]. Let the index im denotes the variables of the instantaneous model, so: Fint = K (x0 − xim ) + B(x0 − xim ) − Mxim
(3)
The instantaneous model (3) is integrated using the initial conditions in (2) to predict the desired velocity in a future time t + ∆t : x im (t + ∆t ) = x im (t ) +
(4)
∫
t + ∆t
t
3.1 Instantaneous model impedance control
nominal position and velocity, and xm , xm , xm are the position, velocity and acceleration of the impedance model. The robot interaction force is measured directly and the end-effector position and velocity are given by forward kinematics:
[
(
) (
The primary advantage of the proposed method is in maintaining the desired impedance relationship in the actual position of the robot, instead of the predicted
The control goal is to cause the end-effector to follow the
x0 , x 0 , x0
Impedance equation
∫
t + ∆t
xim,xim,xim
model
t
Inverse kinematics
qd , qd
+
τ Controller
qr ,qr
Fint
Robot
Internal control loop
xr , xr
Forward kinematics
)]
M −1 − F + B x − x im (τ ) + K x − x im (τ ) dτ int 0 0
qr ,qr
External control loop
Figure 1. Block diagram of instantaneous model impedance control
Fint
position, as in the position-based approach. The integration step ∆t can be chosen to be any time-step. Investigation of its effect on the performance of the impedance controller is beyond the scope of this paper. However, we note that as the gradient of the required acceleration increases, a finer integration step should be used. Hence, it is recommended to chose the integration step as short as physically reasonable, which, in the robotic case is identical to the control cycle time dt .
The control algorithm compensates for the robot dynamics, and adds a correction force proportional to position and velocity errors: (9) τ = − J T Fint + h(q r , q r ) + H (q r )[q d + D(q d − q r ) + P(q d − q r )]
xim (t + ∆t ) =
[
(
) (
The transformation of the desired position, velocity and acceleration to the joint space requires the inverse kinematics formulation L−1 (⋅) and the inverse Jacobian matrix J −1 .
q d = L−1 (xim (t + ∆t )) q d = J −1 xim (t + ∆t )
(6)
(
q d = J −1 x im (t + ∆t ) − Jq d
where the index
d
)
denotes the desired variables. The last
relationship is obtained by differentiating the second relationship. Both the inverse Jacobian and its derivative, which are required to compute the desired joint space position vector in (6), are functions of the desired joint position and velocity. (7)
)]
M −1 − F (t ) + B x − xim (t + ∆t ) + K x − xim (t + ∆t ) int 0 0
J −1 = f (q d )
J = g (q d , q d )
It is noted that the position controller in the inner control loop may not require the entire desired position vector. This is advantageous when the above transformations are not available (as in robots with singular Jacobian matrix). Finally the position controller in the inner control loop generates the control force to achieve the desired position vector. In the present work we use a PD plus inverse dynamic controller so a model of the robot dynamics is needed. In general, the dynamic model of the robot is given by: (8) H (q )q + h(q ,q ) = τ + Tint
3.2 Critical Comparison
Similarly, the desired position is computed by double integration, subject to the initial condition specified in (2). If the inner control loop requires the desired acceleration, it can be isolated from the impedance equation (3) with the new values of position and velocity. (5)
Although both the dynamic-based and the position-based methods were improved during the years, their basic characteristics remain the same. The dynamic-based method depends strongly on the accuracy of the dynamic model of the robot, and, lacking any mechanism for error compensation, is highly sensitive to model uncertainties. Furthermore, the resulting control has a special structure that impedes its use on industrial robots. The position-based method is more robust to uncertainties in robot parameters since it includes compensation mechanisms for position and velocity errors. However, it introduces impedance errors due to discrepancies between the actual robot position and the model position. Specifically, when position errors develop, the measured feedback force acts on a virtual model that is at a different state than the actual state of the robot. Furthermore, the high gains in the position controller, which are required for position tracking, introduce excessive stiffness. The proposed instantaneous model method is designed to overcome the sensitivity of the dynamic-based method to model uncertainties, and to improve the impedance tracking of the position-based method. It does not depend inherently on an accurate model of the robot dynamics. As in the position-based method, the dynamic model may still be used for dynamic compensation at the inner position controller, but it is complemented by the error correction capabilities of the PD controller. The instantaneous model is re-initialized to the current state of the robot so it does not accumulate position errors. Consequently, the discrepancies in the point of application of the measured force feedback are small and the method does not add excessive stiffness to the robot behavior. Moreover, by changing the parameters of the inner loop controller, one can trade between high robustness to model uncertainties and accurate impedance tracking. 4. Simulation 4.1 Robot Task To study the contact stability [3] achieved by the different control methods, we consider a simple task, which involves contact with a rigid wall. The task is
similar to the one described in Hogan [5], but involves a simpler path, which is easier to plan but more challenging to follow. The task includes approaching and making contact with a rigid wall; moving along a section of the wall while maintaining contact; detaching from the wall and returning to the initial position. For simplification, the task is planar and the robot has 2links, as shown in Figure 2. The virtual trajectory ( x 0 , x 0 ) is simply defined. Along
the x-axis the trajectory is composed of two positions, one outside the wall for half of the motion cycle and the cycle. Along the y-axis the virtual trajectory moves up and down, as shown in Figure 2, at constant velocity.
WallWall
Robot path
y Robot
Ke
x
Figure 2. Robot task in 2D environment
Robot links parameters is: l1 = 40[cm ] ; l2 = 30[cm ] ; m1 = 8[kg ] ; m1 = 5[kg ] ; mend = 4[kg ]
The wall contact stiffness is K e = 10 5 [N m ] . The desired impedance equation is: 0 x 62500 0 x F1 100 0 x 3500 0 100 y + 0 y + 0 y = F 3500 62500 2
4.2 Position based - excessive stiffness. This series of simulations demonstrates the effect of the stiffness of the environment on the performance of the controlled robot. The desired impedance model was designed based on the assumed nominal stiffness of the wall. In Figures 4,5, and 6 we present the performance of the different controllers when the environment stiffness is double the nominal one. The effect on the performance of the model is negligible, so we should expect that with good impedance control the robot should perform well. Figure 4 demonstrates that the IM-IC successfully controls the robot and the effect on its performance is negligible too. However doubling the wall stiffness had a severe effect on the robot when the PB-IC is used, as can be seen in Figure 5. Upon making contact with the wall, the robot starts to oscillate, alternately making and breaking contact. This phenomenon is known as contact instability [3], and is one of the main problems that motivated the development of impedance control. When the DB-IC is used, the controlled robot exhibits a relatively long relaxation period, as can be seen in Figure 6. This phenomenon can be explained by the discrete nature of the controller. Due to the large impact forces the gradient of the desired acceleration is large, and so the control frequency is not sufficient to follow the desired impedance accurately. We note that the ability of the IM-IC to overcome this problem stems from its additional feed-forward compensation.
4.3 Dynamic based - sensitivity to uncertainties
Simulation parameters:
proposed method is tested under the same scenarios to study its ability to overcome these difficulties.
The performance of the impedance model is shown in Figure 3. As desired, the model contacts the wall and quickly stabilizes on the desired contact force with no contact instability. In the following sections, we present simulation results with different controllers under two scenarios. Each scenario is designed to reveal the shortcoming of one of the two previous methods. The performance of the
In Figure 7 and 8 we present the performance of the IMIC and DB-IC controllers when the actual load is double the estimated load. Load uncertainties are common, but we note that their effect is not mild. Naturally there is no effect on the model behavior since the desired impedance is the same. Figure 7 demonstrates that the IM-IC successfully controls the robot and the effect on its performance is not severe. However, the robot looses its stability when the DB-IC is used, as can be seen in Figure 8. The lost of stability is attributed to the large position errors that develop upon contacting the wall. This error reflects an impedance error that develops when the interaction forces are large due to the error in the robot model. The ability of the IM-IC to perform well despite model uncertainties stems from its error correction capabilities that alleviates its dependency on the robot dynamic model. Furthermore, a more robust controller, like a P.I.D. controller with no dynamic compensation, may be
incorporated in the inner loop of the IM-IC. Such a controller will improve the robustness to parameter uncertainties but will degrade the nominal performances.
5. Discussion This paper presents a new, practical and accurate method for impedance control. The proposed instantaneous model impedance control provides a solution to the practical problem of uncertainties in the dynamic model of the robot and its environment. The basic method for implementing impedance control, the dynamic-based approach, is highly sensitive to model inaccuracies. The alternative method, the position-based approach, introduces impedance error that appears as excessive stiffness, thereby degrading its performance as an impedance controller. The new method (IM-IC) tracks the desired impedance well even in the face of model inaccuracies. This is due to the instantaneous model prediction mechanism that considers the actual robot position. Thus, it makes it feasible to implement the impedance control strategy in robots with unknown, or poorly specified dynamic model. The proposed method may incorporate a wide variety of inner controllers, which may be tailored by the user for the specific robotic system. Moreover, by tuning the parameters of the inner controller, the user can trade robustness to uncertainties with nominal performance in terms of impedance tracking. A detailed investigation of the IM-IC method for one-dimensional manipulator facilitates precise characterization of its performance in terms of pole placement and supports the conclusions presented here [10] [12]. In summary, the IM-IC method provides an appealing method for implementing impedance control when the dynamic model of the robot or the environment is not reliable or when the control frequency is moderate.
6. Acknowledgement This research was supported by the Promotion of Research at the Technion.
7. References [1] Bonitz.R.G, Hsia.T.C, "Internal Force -Based Impedance Control for Cooperating Manipulators", IEEE Transactions on Robotics and Automation, Vol.12, no. 1, pp. 78-79, Feb. 1996. [2] Colbaugh.R, Seraji.H, Glass.K, Direct Adaptive Impedance Control of Robot Manipulators. Journal of Robotic Systems 10(2), 217-248 (1993). [3] Colgate .J.E and Hogan.N, An analysis of contact instability in terms of passive physical equivalents, in
Proc. IEEE Int. Conf. Robotics and Automation, 1989, pp. 404-409. [4] Hogan, N. 1985, Impedance Control: An Approach to Manipulation. Part I-III. Journal of dynamic System, Measurement' and Control' vol.107. [5] Hogan.N, stable execution of contact tasks using impedance control. in Proc, IEEE Int. Conf. Robotics and Automation, Raleigh, NC, Mar. 1987, pp. 10471054. [6] Jung.S, Hsia.T.C, Bonitz R.G, On robust impedance force control of robot manipulators. Procceecings-IEEE International Conference on Robotics and Automation, v3 1997, IEEE, Piscataway, NJ, USA. p 2057-2062. [7] Lawrence.D.A, Impedance Control Stability Properties in Common Implementation. Proc. IEEE Int. Conf. on Robotics and Automation, pp. 1185-1190. 1988. [8] Lu.W.S, Meng.Q.H, Impedance Control with Adaptation for Robotic Manipulations. IEEE transaction on robotics and Automation, vol 7 no. 3 June 1991.
[9] Newman.S.W, Branicky.S.M, Poddgurski.A.H, Force-Responsive Robotic Assembly of Transmission Components Proceedings of the 1999 IEEE. International Conference on Robotics and Automation. May 1999, pp. 0296-2102. [10] urdilovic.D, Synthesis of Impedance Control Laws at Higher Control Levels: Algorithms and Experiments. Proceedings of the IEEE, International Conference on Robotics and Automation, Leuven, Belgium, May 1998. [11] Valency T., Instantaneous Model Impedance Control for Robots. M.S.c thesis. Technion -Israel Institute of Technology, Haifa August 1999. [12] Valency T, Zacksenhouse M, Instantaneous Model Impedance Control for Robots. In preparations.
Model position
Wall PB-IM position
Desired position
Desired position
Figure 3. Performance of the impedance model.
Figure 6. Performance of the robot under DB-IC when the environment stiffness is double the nominal one.
X0 Model Alone IM-IC
IM-IC position
Desired position
Figure 4. Performance of the robot under IM-IC when the environment stiffness is double the nominal one.
Desired position
Figure 7. Performance of the robot under IM-IC when the estimated load is half the real load.
X0 Model Alone DB-IC
PB-IC position
DB-IC position
Desired position
Figure 5. Performance of the robot under PB-IC when the environment stiffness is double the nominal one.
Desired position
Figure 8. Performance of the robot under DB-IC when the estimated load is half the real load.
