Contact Stiffness and Damping Estimation for Robotic Systems

Transcription

1 D. Erickson M. Weber I. Sharf Department of Mechanical Engineering McGill University Montreal, PQ Canada Contact Stiffness and Damping Estimation for Robotic Systems Abstract In this paper, we review and compare four algorithms for the identification of contact stiffness and damping during robot constrained motion. The intended application is dynamics modeling and simulation of robotic assembly operations in space. Accurate simulation of these tasks requires contact dynamics models, which in turn use contact stiffness and damping to calculate contact forces. Hence, our primary interest in identifying contact parameters stems from their use as inputs to simulation software with contact dynamics capability. Estimates of environmental stiffness and damping are also valuable for force tracking and stability of impedance controllers. The algorithms considered in this work include: a signal processing method, an indirect adaptive controller with modifications to identify environment damping, a model reference adaptive controller and a recursive least-squares estimation technique. The last three methods have been proposed for real-time implementation in impedance and force-tracking controllers. The signal processing scheme uses a frequency estimate calculated with fast Fourier transform of the force signal and is an off-line method. The algorithms are first evaluated using numerical simulation of a benchmark test. Experiments conducted with a robotic arm contacting a flexible wall provide a further demonstration of their performance. Our results indicate that the indirect adaptive controller has the best combination of performance and ease of use. In addition, the effect of persistently exciting signals is discussed. KEY WORDS contact parameters, identification, estimation, contact dynamics, environment stiffness and damping, impedance control, adaptive control, recursive least-squares 1. Introduction Robotic tasks can be classified into two categories: unconstrained and constrained motion. Unconstrained motion occurs when the manipulator is instantaneously free to move in The International Journal of Robotics Research Vol. 22, No. 1, January 23, pp , 23 Sage Publications any direction without contacting the environment. Examples of such tasks include spray painting and visual inspection. Constrained motion occurs when the manipulator interacts with its environment through a point or multiple points of contact. Tasks such as grinding, cutting and assembly demonstrate constrained motion. Modeling and control of constrained robotic operations present a number of challenges. Accurate simulation of these tasks requires the contact dynamics capability in formulating and solving the motion equations. From the control standpoint, it is widely acknowledged that a force control or combined position/force control strategy is required to ensure desired interaction with the environment. One of the most popular control strategies for constrained tasks is impedance control (Hogan 1985) Impedance Control Impedance control utilizes a single control law which attempts to regulate both position and force by specifying a dynamic relationship between them. This relationship is chosen to be a second-order linear impedance because such systems are well understood and simple to control. A standard impedance control law is shown in eq. (1), where M t, B t, K t are the target impedance mass, damping and stiffness, x r is the reference trajectory, x is the actual trajectory of the end-effector, and F e is the external force applied to the end-effector: M t (ẍ ẍ r ) + B t (ẋ ẋ r ) + K t (x x r ) = F e. (1) The target impedance parameters are specified by the user and have the dimensions of the task space. They cause the manipulator to exhibit the dynamics of a multi-directional mass spring damper system. The target impedance matrices are typically chosen to be diagonal, resulting in uncoupled response along each principle direction of x. 41

2 42 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / January Environment Model For the purposes of the present paper, it is necessary to distinguish two approaches to environment modeling. The first is used for impedance and force control and it represents the environment as an n-dimensional spring (n 6), or a spring and damper, generally referred to as impedance. Thus, when the manipulator contacts the environment, a reaction force results which can be defined as F e = K e (x e x) + B e (ẋ e ẋ). (2) Here, K e and B e are n-dimensional matrices representing the stiffness and damping characteristics of environment, while x e represents the environment location. It is usual, however, to treat each Cartesian variable independently or, in other words, to assume that the environment impedances in different directions are uncoupled (Seraji and Colbaugh 1997; Singh and Popa 1995). In this case, representation (2) is replaced by n scalar equations of the form: F e = K e (x e x) + B e (ẋ e ẋ). (3) This one-dimensional version of the environment model is shown schematically in Figure 1, where x e, K e and B e are scalars representing the environment location before contact, stiffness and damping, respectively. Implicit in the models of eqs. (2) and (3) is the fact that the interaction between the robot and the environment is confined to a single point or a small region, such as the case of robotic grinding operations. Other common applications of this model are for robot collisions with a wall or motion along the wall. Note that some researchers also include environment inertia in their models (missing in our eq. (2)). One of the identification methods explored in this paper (Love and Book 1995) can directly estimate the inertia of the environment. For many applications, such as space, it is reasonable to assume that the inertia of the environment is known. In terrestrial applications, environments tend to be stationary or quasi-static, in which case the inertia term can be neglected. In the context of contact dynamics modeling, the robot environment interaction is distributed over a finite number of contact points between the robot payload and the fixture. At each point, a contact force can be defined in terms of its normal and tangential components as F c = F n n + F t t. (4) One common model used to define the normal component of the contact force is the spring-dashpot model (Gilardi and Sharf 22) of the form: F n = K e δ + B e δ. (5) In the above, the contact parameters, stiffness K e and damping B e, relate the local deformation δ and its rate to the normal Fig. 1. Environment model. force at each contact point between the contacting objects. In the context of contact dynamics modeling, δ at each contact point can be calculated from the geometry and location of the two mating objects. Note that we have retained the subscript e in denoting contact parameters for uniformity of notation. In the simplest contact scenario, that of one-point contact between robot end-effector and fixture, F c of eq. (4) becomes the external end-effector force F e and the contact model in the normal direction (eq. (5)) is completely analogous to the onedimensional environment model of eq. (3) with δ = x e x. In this case, the problems of contact parameter and environment impedance estimation are identical. For general threedimensional contacts with multiple points of contact between the payload and the fixture, and assuming identical contact parameters at all contact points, the net contact force can be written as the sum of individual contact forces N N N F e = K e δ i n i + B e δ i n i + F ti t i, (6) i=1 where N is the number of contact points and F ti is determined by the particular friction model. It is noted that the contact parameters employed in eqs. (5) and (6) are by definition scalar quantities, regardless of the complexity of contact geometry. For both representations of the environment, in general, the stiffness and damping parameters are poorly known, while environment location or the contact geometry are either known or can be determined. This implies that the parameter estimation problem can be formulated as a linear identification problem of the form F e = φ T θ, (7) where θ contains the contact parameters to be estimated and φ, called the regressor, is strictly a function of contact geometry. i=1 i=1

3 Erickson, Weber, and Sharf / Contact Stiffness and Damping Estimation 43 The above is true for one-point (one-dimensional) contact or for general three-dimensional scenarios with multiple points of contact. In the former case, the regressor is φ =[x e x,ẋ e ẋ] T (or [δ, δ] T ) and the vector of parameters is θ =[K e,b e ] T. The reader is referred to Weber et al. (22) for derivation of eq. (7) for complex contact geometries. In this case, the main complication is that friction identification is now coupled with contact stiffness and damping estimation, thus making the solution prone to approximations and assumptions of friction modeling Motivation For certain tasks such as space-based manipulation or nuclear waste remediation, the need for high-fidelity simulation is well understood. To simulate robot environment contact tasks, the simulation software must incorporate a contact dynamics model, such as in eq. (5), and apply appropriate solution methodology. One such software package is MDSF (Ma et al. 1997) developed by MacDonald Dettwiler Space and Advanced Robotics, Ltd. 1 A recently completed experimental validation of MDSF s contact dynamics capability (Van Vliet et al. 2) has demonstrated that for high-fidelity simulation, good estimates of contact parameters must be known. The identification of environment characteristics is also an integral component of force-tracking impedance controllers. One drawback of position-based impedance control is its inability to directly control the interaction force F e. Instead, the contact force results from the target impedance values, reference trajectory and environment properties. A few researchers have designed force-tracking impedance controllers which command the robot to exert a specified force on the environment. For example, Seraji and Colbaugh (1997) developed direct and indirect adaptive control to achieve force tracking by estimating the environment stiffness and location. Performance was demonstrated in one dimension by using a seven-degrees-of-freedom (7-DoF) robot to exert a desired force normal to the reaction surface, while tracking a desired trajectory tangent to the surface. Singh and Popa (1995) also showed that model reference adaptive control (MRAC) could be applied to impedance force control to achieve force tracking. It was found that, like the algorithm of Seraji and Colbaugh, an estimate of the environment stiffness was required. Environment estimation can also be used to improve stability during contact. Love and Book (1995) showed how estimates of environment impedance parameters could be used to design an impedance controller that was stable both during and at the onset of contact. They also showed that, in the absence of accurate estimates, unstable contact could result. Their algorithm uses a recursive least-squares () technique applied to a discrete form of the environment dynamics equations proposed by An et al. (1988). 1. Formerly SPAR Aerospace, Robotics Division About This Paper In this paper we compare the existing environment parameter estimation algorithms (Seraji and Colbaugh 1997; Singh and Popa 1995; Love and Book 1995), as well as an original method which uses a signal processing approach, all applied to the problem of contact or environment stiffness and damping estimation. We outline some important changes made to improve the accuracy of parameter estimation of the indirect adaptive controller. Our comparison is based both on numerical performance of the methods and implementation on a real system. In addition, an important aspect of parameter estimation the requirement for persistent excitation of the reference signal is discussed and demonstrated experimentally. Following other authors, our brief presentation of the different methods is limited to one-dimensional (for the environment impedance model) or one-point contact (in the case of contact model) situations. It is emphasized, however, that although the bench test considered in this paper represents a simple contact geometry single-point, onedirectional contact our findings have direct relevance to multiple-point contacts for complex mating bodies. As noted earlier, the identification problem for such scenarios can still be formulated as a linear relationship between the net contact (end-effector) force and the scalar contact parameters to be identified. With the exception of the signal processing technique, the other methods investigated here can be directly applied to identify contact parameters for three-dimensional contact geometries, using appropriate definitions of regressor φ and parameter vector θ. 2. Environment Estimation Algorithms 2.1. Signal Processing Method The signal processing method of environment estimation was developed from the theory of second-order, linear, time invariant systems. A robot that is controlled using impedance control exhibits a second-order dynamic relationship between the position of the end-effector and the external force applied to it. The characteristics of this relationship are governed by the target impedance values (M t,b t,k t ), through which the user can specify the dynamic behavior of the manipulator (Hogan 1985). Also recall that the environment is assumed to behave as a linear impedance according to its stiffness and damping coefficients (K e,b e ). When these two systems come into contact, a new second-order system is formed (see Figure 2) which is composed of the impedance characteristics of both the controller and environment. If the impedance characteristics of the combined system (impedance controller and environment) can be determined, then those of the environment can be calculated.

4 44 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / January 23 Fig. 2. Second-order system formed by environment and impedance controlled robot Determination of System Natural Frequency and Damping Ratio The impedance properties of the environment robot system can be determined from the step response of this system. This is achieved by commanding a step in the location of the robot end-effector (x) during contact, and measuring the resultant force (F e ). Assuming an underdamped response, the damped natural frequency ω d can be determined from the fast Fourier transform (FFT) of the force signal. To determine the damping ratio ζ, we have used the settling time method which gives the following relation for T s to achieve convergence to within 5% of the steady-state value e ζωnts =.5, (8) where ω n is the undamped natural frequency (Dorf and Bishop 1995). The settling time is then given by T s = (9) ζω n and the number of cycles of the response before the settling time is reached is #cycles = ζ 2. (1) 2πζ From the above, the damping ratio is obtained as ζ =.4768 #cycles (11) The solution for ζ is therefore found by counting the number of visible cycles (including fractional parts) before the measured contact force response converges to within 5% of the steady-state value, and substituting the result into eq. (11) Determination of Environment Stiffness and Damping Coefficients The stiffness and damping coefficients of the environment can be extracted from the known values of ω d and ζ. For the robot environment system in Figure 2, the equivalent stiffness, damping and mass are K eq = K t + K e (12) B eq = B t + B e (13) M eq = M t. (14) In terms of these parameters, the natural frequency and damping ratio are given by ω n = ω d 1 ζ 2 = K eq M eq (15) B eq ζ = 2. (16) K eq M eq With the values for ω d and ζ determined from the force response and equations (12) (16), we obtain the environment stiffness and damping in terms of the known values: K e = ω 2 M n t K t (17) B e = 2ζ (K t + K e )M t B t. (18)

5 Erickson, Weber, and Sharf / Contact Stiffness and Damping Estimation 45 It is important to note the advantages and disadvantages of this method. To its credit, the signal processing method requires very little sensory data. Only the (normal) contact force needs to be measured and it is common for robots that perform contact tasks to have a force sensor mounted at the end-effector. This algorithm then processes the force measurements off-line, after the experiments are completed. Unlike the other methods described below, this scheme does not require measurements of the environment deflection or velocity. Interestingly, this method can be best described as a hybrid between basic frequency-domain and time-domain techniques, as it makes use of both types of information. One disadvantage is that, in order to measure both ω d and ζ, the force response must be underdamped. This implies that the target impedance values must be carefully chosen to give the desired behavior. Frequency-domain methods in general are not subject to this limitation, but for our problem they require kinematic data (i.e. position or velocity). Also, the method cannot be easily extended to identify contact parameters for general contact geometries because it makes use of the equivalent stiffness and damping for the robot environment system. These cannot be easily defined for three-dimensional contacts with multiple contact points, as in the environment model of eq. (6). Finally, the algorithm assumes that the manipulator dynamics are represented perfectly by the target impedance which is not achievable in practice Adaptive Control The indirect adaptive controller proposed by Seraji and Colbaugh (1997) was intended to achieve force tracking within impedance control. This implies that the measured force (F e ) should converge to a desired reference force (F r ). To achieve this goal, a trajectory generator was developed which modified the reference trajectory (x r ) on-line, during the contact. The algorithm uses estimates of the environment stiffness and location ( ˆK e, ˆx e ) to calculate x r in the impedance control law (1) from x r =ˆx e + 1ˆK e F r. (19) The environment estimates are adapted during each time step according to an update law derived using a Lyapunov-based approach. This method uses the popular regressor form of the environment impedance equations. The results presented by Seraji and Colbaugh (1997) showed that force tracking was achieved, but convergence of the environment estimates to the correct values could not be guaranteed. This was attributed to lack of a persistent excitation condition. Our experience with numerical simulation, however, revealed that the addition of persistent excitation still did not result in parameter convergence. The cause for this is the choice of stiffness and environment location as adjustable parameters. These parameters are multiplied together in the contact dynamics equations (2), which therefore cannot be written in the standard linear in parameters form. In light of this, the indirect adaptive controller was reformulated to estimate environment stiffness and damping ( ˆK e, ˆB e ), but not location ( ˆx e ). This is in line with our previous comments that location of the environment is either known (as may be the case in terrestrial applications) or can be measured, similarly to the location of robot end-effector already required for most methods considered here. With this modification, the actual contact force remains as in eq. (2), the trajectory generator becomes and the estimated force becomes x r = x e + 1ˆK e F r (2) ˆF = ˆK e (x e x) ˆB e ẋ (21) where we make use of constant x e. The force estimate can now be written in the linear form by defining the regressor φ = [x e x, ẋ] T and the vector of parameter estimates ˆθ =[ˆK e, ˆB e ] T ˆF = [ x e x, ẋ ] [ ˆK e ˆB e ] = φ T ˆθ. (22) Subtracting eq. (3) from eq. (22) yields F = φ T θ (23) where F = ˆF F e and θ = ˆθ θ. The adaptation law can be formed using a Lyapunov technique (Slotine and Li 1991) by defining the energy function V, where Ɣ is a positive definite matrix: V = θ T Ɣ θ. (24) If the parameter adaptation law is chosen as θ = Ɣ 1 φ F (25) it can be shown, using eqs. (24) and (25), that the derivative of the energy function is V = 2 θ T φφ T θ (26) which is negative definite in terms of θ provided αi n φφ T dt βi n for <α<β. (27) The above condition on the regressor is in fact the definition of persistent excitation (Anderson et al. 1986). Since eq. (26) is negative definite in terms of θ, it can be concluded that the estimate error θ as t, or that the environment estimates ( ˆK e, ˆB e ) converge to the actual values (K e, B e ).

6 46 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / January 23 If the actual parameter values are assumed to be constant, then θ = ˆθ and the adaptation law (25) can be used to generate new parameter estimates during each time step by numerical integration. The new estimates are then used to update the trajectory generator (2). The resulting impedance controller achieves both force-tracking and accurate parameter estimates, the latter in the presence of persistent excitation. The implementation of the parameter estimation in the indirect adaptive controller requires data on the actual position and velocity of the end-effector (x,ẋ), and the interaction force (F e ). In practice, accurate measurement of absolute kinematics quantities at the robot tip is difficult to achieve and this poses a challenge in the application of the method. To use the algorithm, the user must specify the gain matrix Ɣ for the adaptation law (25) and the reference force signal F r. This force input must be persistently exciting if accurate estimates for both the environment stiffness and damping are required Model Reference Adaptive Control MRAC was used by Singh and Popa (1995) to investigate some fundamental issues of force control, namely explicit force control, general impedance control and force-tracking impedance control. Of main interest to us is the latter since it is the impedance control combined with explicit force control that involves environment parameter estimation. In fact, the work of Singh and Popa demonstrated that to achieve a forcetracking impedance controller, knowledge of the environment contact parameters was required. The MRAC controller presented by Singh and Popa is significantly more complicated than the indirect adaptive controller proposed by Seraji and Colbaugh (1997). The full controller is not described in detail here, but a brief explanation of the method is given below. Similarly to eq. (3) the MRAC controller is based on the model of the environment as a linear spring in parallel with a viscous damper. The major components of the controller include the following: 1. The plant model represents the actual dynamics of the manipulator in contact with environment. It is assumed that the system is linear and the state-space parameters are in general unknown. 2. The reference model represents the target impedance with the reference signal specified by the virtual trajectory generator. 3. The MRAC parameter estimator provides the adaptation law for system parameters that match the plant model to the reference model. 4. The environment parameter estimator provides the adaptation law for the environment stiffness and damping. The state dynamics are driven by the model-following control law which involves state feedback (x,ẋ) and an appropriately chosen auxiliary signal. The virtual trajectory dynamics are specified by the user, and this signal must be persistently exciting to ensure convergence of the model-matching parameters (Singh and Popa 1995). Furthermore, it is stated that environment parameter convergence can be guaranteed if the desired contact force is time-varying (persistently exciting). The experimental results presented by Singh and Popa show accurate estimation of the environment stiffness and damping for a variety of environments. Simultaneously, a desired contact force is achieved Recursive Least Squares Love and Book (1995) have successfully demonstrated that contact stability can be improved if estimates of the environment impedance parameters are known. They modeled the environment as a locally stationary mass spring damper system: F e = M e ẍ + B e ẋ + K e (x x e ). (28) By defining δx = x x e and remembering that the environment is stationary (ẍ e =ẋ e = ), eq. (28) can be rewritten as F e = M e δẍ + B e δẋ + K e δx. (29) The above differs slightly from the equation used by Love and Book (1995), but instead reflects the original work described by An et al. (1988). The bilinear transformation is then used to transform eq. (29) into its discrete-time counterpart [ ( ) 2 ( 2 1 z 1 ) 2 F e = M e T 1 + z 1 ( )( 2 1 z 1 ) ] (3) + B e + K T 1 + z 1 e δx where T is the sampling period. We can expand eq. (3) into a polynomial in terms of z. By recognizing that z 1 represents a shift of one step in the time domain, and letting k describe the time-step index, the corresponding difference equation is F e[k] + 2F e[k 1] + F e[k 2] [ ( ) 2 ( ) ] 2 2 = M e + B e + K e δx [k] T T [ ( ) 2 ] K e M e δx [k 1] T [ ( ) 2 ( ) ] M e B e + K e δx [k 2] T T = Aδx [k] + Bδx [k 1] + Cδx [k 2] (31)

7 Erickson, Weber, and Sharf / Contact Stiffness and Damping Estimation 47 which can be cast into the regressor form as with y [k] = φ T [k] θ [k] (32) y [k] = F e[k] + 2F e[k 1] + F e[k 2] (33) x [k] x e φ [k] = x [k 1] x e x [k 2] x e (34) θ [k] = [ A B C ] T. (35) The solution for θ [k] takes the following form (Ljung 1987) ( θ [k] = θ [k 1] + L [k] y[k] φ T θ ) [k] [k 1] (36) Fig. 3. Manipulator and flexible wall used for the benchmark test. where P [k 1] φ [k] L [k] = (37) λ + φ[k]p T [k 1] φ [k] P [k] = 1 ( P [k 1] P [k 1]φ [k] φ T P ) [k] [k 1]. (38) λ λ + φ[k]p T [k 1] φ [k] The initial guess for the adaptation gain matrix P and the weighting factor λ( λ 1) must be specified by the user. Once θ [k] has been calculated (and subsequently A, B, C), the desired parameter estimates at the kth sample period can be recovered using M e = 1 4 ( T 2 ) 2 (A + C B), B e = ( ) T (A C), 4 K e = 1 (A + B + C). (39) 4 Similarly to the indirect adaptive scheme, the least-squares algorithm requires measurements of contact force F e and the corresponding end-effector position x. It is noted that unlike the other two methods considered here, the solution is geared to identifying impedance parameters within the framework of pure impedance control, without tracking a desired force signal. Formulated as a linear identification problem, the success of the parameter estimation is subject to the standard persistent excitation condition as stated in eq. (27). 3. Description of Benchmark Test 3.1. Physical Setup To compare the four schemes presented above, a benchmark test was developed. This test was analyzed using both a numerical simulation in MATLAB, as well as experiments conducted with the Planar Robotics Testbed at the University of Victoria. The testbed includes a custom manipulator assembled using modular links and joint motors, as well as a flexible wall fixture (see Figure 3). Fig. 4. Detail of flexible wall fixture. The flexible wall represents a relatively simple environment designed specifically for our experiments as a first step to evaluating the performance of different techniques. In particular, the flexible wall is attached to a set of linear bearings, and can thus translate in one direction. Behind the wall are a set of coil springs and oil-filled dampers. These components are also modular, and can be added or removed to change the mechanical impedance of the wall (see Figure 4). For the benchmark test presented here, the stiffness of the environment was defined by the two springs in parallel, with the effective stiffness of K e = 48 N m 1. This represents a relatively soft environment the choice made partially because of the limited accuracy of the wall displacement sensor. In practical applications, we encounter much stiffer environments which makes the identification task more challenging. The damping coefficients of the dampers are not accurately known but are estimated to give approximate environment damping of B e = 2 kg s 1.

8 48 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / January 23 x r (m) Reference trajectory Flexible wall (K e,b e ), only one non-zero frequency is required (Landau et al. 1998). Therefore, a persistently exciting input to the MRAC and indirect adaptive schemes was defined by adding a small-amplitude sinusoid to F d to give F r = F d + 8 sin(2t). (4) For the scheme, the persistent excitation must be added directly to the desired trajectory, giving.2 x r = x d +.2 sin(2t), (41) Fig. 5. Trajectory of manipulator, showing contact with flexible wall. Instrumentation on the testbed includes: joint encoders on the manipulator, a planar force/torque sensor mounted at the end-effector and a linear displacement potentiometer directly connected to the flexible wall. The force sensor measures the contact force at the robot end-effector. The potentiometer gives a measurement of the wall deflection from which the end-effector velocity is calculated using a finite difference scheme (for the indirect adaptive and MRAC methods). We note that, although end-effector force sensors are common on robot manipulators, an absolute position sensor capable of measuring the environment deflection in three dimensions may be difficult to implement. This is especially true for stiff environments Trajectory and Inputs The manipulator was commanded to follow a reference trajectory using impedance control. The reference trajectory used in numerical simulations is shown in Figure 5 and was defined by a fifth-order polynomial for the approach and withdrawal phases of the maneuver. Part of the trajectory lies inside the flexible wall, resulting in robot environment contact for several seconds. Note that during the contact phase, a desired force (F d = 5 N) was specified for the indirect adaptive and MRAC controllers, with the reference trajectory modified accordingly (see eq. (2) for the indirect adaptive controller). As discussed earlier, accurate damping estimation can be guaranteed only with a persistently exciting reference signal. A common approach to create such a signal is by adding several sinusoidal signals with distinct frequencies. It has been shown that the number of required frequencies in the reference signal depends on the number of parameters to be determined. For a system involving two unknown parameters where x d is the desired (reference) trajectory before the addition of persistent excitation. The amplitudes of these signals were chosen to give a good signal-to-noise ratio, while remaining within the physical capabilities of the manipulator and environment. The frequency of the sinusoid is sufficiently far from zero while still within the bandwidth of the manipulator. When the choice of the excitation frequency is not evident or to increase the temporal bandwidth of the excitation, a pseudo-random binary sequence (PRBS) can be used. This signal resembles a square wave with a constant amplitude, but has a randomly varying period (Landau et al. 1998). As such, it has a constant spectral density over a broad bandwidth of frequencies. Experimental results obtained with PRBS excitation are presented in Section 5. The target impedance coefficients used in the indirect adaptive and algorithms were chosen to be: M t = 5 kg, B t = 1 kg s 1 and K t = 5 N m 1. These nominally cause the manipulator-environment system to have a natural frequency of ω n = 1 rad s 1 and critical damping. The MRAC algorithm was found to be unstable during contact when these target impedance values were used, so this algorithm was individually tuned, resulting in: M t = 1 kg, B t = 65 kg s 1 and K t = 1e5Nm 1. Finally, to produce an underdamped response for the signal processing method, we used: M t = 2 kg, B t = 5 kg s 1 and K t = 2 N m 1. Each algorithm was given the same initial conditions for the parameter estimates: ˆK e[] = 3 N m 1 and ˆB e[] = 5 kg s 1. Other initial guesses were also attempted with, overall, not significantly different outcomes. Note that, in practice, the initial guess for environmental stiffness should error on the soft side in order to avoid excessive transients in the contact forces. The adaptation gain matrix used in the indirect and MRAC schemes was chosen through trial and error to be Ɣ = diag{5.e4, 2.5e3}. The scheme required an initial value for P [] = γi n, where γ is a large constant (Landau et al. 1998). For our simulations and experiments we set P [] = 1 5 I n. The weighting factor (λ), which is often chosen to be slightly less than one, was set to λ =.98 (Landau et al. 1998). Using these values, simulation and experimental results of the benchmark test were obtained both with and without persistent excitation, as presented below.

9 Erickson, Weber, and Sharf / Contact Stiffness and Damping Estimation Simulation Results The benchmark test was first implemented as a numerical simulation using MATLAB. To model the noise inherent in the force sensor used in our experiments, a random signal with an amplitude of ±1 N was added to the calculated contact force. On the other hand, perfect end-effector position and velocity measurements were assumed in the simulations Results With Persistent Excitation Sinusoidal excitation was added to the reference signal of the indirect adaptive, MRAC and algorithms (recall that the signal processing method uses a trajectory with a step input, so this algorithm should not be used with persistent excitation). For these three methods, the prescribed trajectory caused the manipulator to be in contact with the wall for just over 3 s, from t = 5.9 stot = 9.2 s. During this time, all three algorithms showed a definite parameter adaptation of the stiffness and damping estimates towards the actual values (see Figure 6). The stiffness plot showed that the indirect adaptive controller and the MRAC controller resulted in a very similar response. In particular, both methods converged within 1 s and produced stiffness estimates with less than.1% error (as defined by the value at the last instant of contact). Although the algorithm demonstrated significant overshoot, convergence was also rapid and accurate, with a final stiffness estimate error of 1.3%. The plot of damping estimation demonstrates good convergence for each algorithm (see the lower plot in Figure 6). The response was, in general, rapid with little overshoot. In addition, all algorithms resulted in damping parameter estimates with less than 1.% error. Through many simulations, it was apparent that the damping estimation was dramatically improved if the persistent excitation signal was increased (either in amplitude or frequency). This is reasonable since either change would result in faster manipulator reference velocities (ẋ r ) and therefore more rapid convergence to the actual parameter value. Based on these simulations, it can be concluded that in the presence of persistent excitation all three algorithms are capable of very accurate stiffness and damping estimation. It is noted that the rate of parameter convergence is highly dependent on the choice of adaptation gains. These gains are chosen by the user (usually by trial and error) which implies that the parameter convergence rate can be controlled to some extent. The indirect adaptive controller and MRAC controller use identical parameter adaptation laws and gain matrices, allowing for direct comparisons of these algorithms. The scheme, however, is fundamentally different in the manner in which it forms parameter estimates and therefore cannot be directly compared to the other schemes Results Without Persistent Excitation For practical applications, persistent excitation may prove to be difficult or hazardous. For stiff environments, the amplitude of possible motions may be too small to have a positive effect on convergence, while for fragile environments the time varying force signal may cause damage. Therefore, it was deemed important to study how the contact parameter estimation algorithms perform in the absence of persistent excitation. The same benchmark test was performed, but the superimposed sinusoidal signal was removed from F r or x r in eqs. (??) and (??). The measured force response for the signal processing method during contact is given in Figure 7. The contact force demonstrates the desired second-order underdamped behavior, as shown. For this simulation, the response converges to within 5% of the steady-state value in approximately 1.5 cycles, which results in a calculated damping ratio of ζ =.33. The frequency response of this signal shows a single dominant frequency at ω d = 2.93 Hz (see Figure 8) which corresponds to the undamped natural frequency ω n = 3.7 Hz. Following the method outlined in Section 2, the stiffness and damping estimates were found to be 5441 N m 1 and 173 kg s 1, resulting in a parameter error of 13% for both values. Results from the indirect adaptive, MRAC and methods are given in Figure 9, and demonstrate some interesting trends. The stiffness estimates still converged to nearly the correct value, with errors for the indirect, MRAC and schemes equal to 2.4%,.1% and 1.7% respectively. As before, the algorithm demonstrated significant overshoot, but converged rapidly to nearly the actual value. The response of the indirect and MRAC controllers was smooth, rapid and accurate. Thus, the absence of persistent excitation does not significantly affect the stiffness estimation when applied to simulated results. Substantial degradation, however, is apparent in the damping estimates. As can be seen from the damping (lower) plot in Figure 9, the estimate converged slowly, and exhibited a large amount of noise in the response. The contact phase was sufficiently long however (3.3 s) to allow for estimation of damping to occur with the final estimate error of 2.5%. Thus, the algorithm demonstrates damping convergence even in the absence of the sinusoidal persistent excitation signal because the trajectory is slowly time-varying. For the indirect adaptive and MRAC schemes, it is evident that adaptation occurs exclusively during the transient phases of the maneuver: the onset and conclusion of contact. Due to the force-tracking nature of these algorithms, it is only during these times that the manipulator has a non-zero velocity normal to the wall. As evidenced by the accurate estimates of B e calculated with the indirect and MRAC algorithms (3.% and.% errors, respectively), adaptation during these brief moments is possible. However, the convergence rate is highly dependent on the adaptation gains, and thus convergence to actual values

10 5 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / January Stiffness (N/m) MRAC Damping (kg/s) 2 1 MRAC Fig. 6. Simulation results of stiffness and damping estimation with sinusoidal excitation. Force (N) Calculated force 5% bounds Magnitude Fig. 7. Simulated force response during contact for the signal processing method Frequency (Hz) Fig. 8. Frequency response of contact force signal.

11 Erickson, Weber, and Sharf / Contact Stiffness and Damping Estimation Stiffness (N/m) MRAC Damping (kg/s) 2 1 MRAC Fig. 9. Simulation results of stiffness and damping estimation without persistent excitation. cannot be guaranteed before the velocity of the contact point approached zero. 5. Experimental Results The simulation results presented above indicate that all four algorithms are capable of accurate environment parameter estimation. Using the signal processing method, acceptable stiffness and damping estimation were possible without persistent excitation. For the indirect adaptive, MRAC and algorithms, stiffness estimation is very accurate either in the presence or absence of persistent excitation, but reliable damping estimation cannot be guaranteed without additional excitation. Experimental validation of the indirect adaptive, and signal processing algorithms was performed by implementing the benchmark test on the Planar Robotics Testbed described in Section 3. As demonstrated in simulations of Section 4, the MRAC algorithm generated very similar parameter estimates and responses to those of indirect adaptive method since the two share identical parameter adaptation laws and gain matrices. At the same time, the indirect adaptive controller was found to be intuitive and easily implemented, while the MRAC method was considerably more complex. For these reasons, we concluded that the MRAC algorithm offers no significant advantage over the indirect adaptive controller and therefore it was not investigated with the Robotics Testbed Data Collection and Processing All experiments were performed with a sampling rate of 5 Hz. The measured data were filtered by using the Butterworth filter of second order with cut-off frequency set to 5 Hz. These filter parameters were optimized manually to minimize the errors in stiffness and damping estimates. The speed of the wall was calculated by using the finite-difference scheme on the wall deflection and filtering the result. Finally, some static friction is present at the bearings of the wall fixture and it was determined that a force of 6Nisrequired to move the wall from its stationary position. This value was subtracted from the force measurements to give the contact force employed in the estimation algorithms Results With Persistent Excitation adaptive and parameter estimation results are presented for the sinusoidal and PRBS excitations in the input signal. In each case, 1 experiments were performed to

12 52 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / January x r (m).3.2 IA trajectory.1 Wall trajectory Fig. 1. Reference trajectories used for the indirect adaptive and experiments with sinusoidal excitation. arrive at the average parameter estimates reported below. The contact force and wall deflection profiles measured in one of the tests with sinusoidal excitation are shown in Figure 11. The reference trajectories used for the and indirect adaptive experiments with sinusoidal excitation are plotted in Figure 1. These were designed to produce comparable deflections and forces for a more truthful evaluation of the and indirect adaptive estimation results. The corresponding profiles of the stiffness and damping estimation are illustrated in Figure 12. Stiffness estimation for both algorithms is fairly good, with errors of 12.6% for the indirect adaptive controller and 3.5% for the technique. (Note that, as in simulation results, these errors are calculated for the estimates at the last instant of contact.) Convergence was rapid in each case (5 1 periods of oscillation), with the indirect adaptive controller exhibiting a smooth response while the scheme again showed a large initial overshoot. The estimate also trailed during the withdrawal from the wall. These differences in performance concur with the differences in trajectories for the two maneuvers as can be seen in Figure 1. Damping estimation does not exhibit clear convergence, although the end-of-contact values are reasonable for both methods. The average errors for damping are 15.7% and 22.% for the indirect adaptive and controllers, respectively. To verify convergence of damping estimation, experiments were conducted with a longer contact maneuver. The corresponding results are shown in Figures 13 and 14. These reconfirm convergence of the stiffness estimates observed before, and demonstrate convergence of damping albeit to a value higher than expected. A set of results obtained by using PRBS excitation described in Section 3.2 is included in Figure 15. The excitation signal was generated by using 1 binary registers which were shifted at a frequency of 2 Hz. The feedback signal was created from a combination of the 7th and 1th register values (as suggested in Landau et al. 1998). The resulting signal, composed of zeros and ones, was scaled and shifted to produce an excitation signals of ±8 N and ±.2 m for the force and motion inputs, respectively. The results in Figure 15 exhibit similar characteristics to those obtained with the sinusoidal excitation (Figure 12), but with a noisier convergence. Corresponding results for the longer maneuver are shown in Figure 16. These exhibit similar features to PRBS results in Figure 15, but with a more definitive convergence of estimation Results Without Persistent Excitation Experiments without the addition of an excitation signal were performed with the signal processing, indirect adaptive and methods. As before, 1 experiments were carried out for each procedure to generate an average stiffness and damping estimate. The values obtained with the signal processing method were 329 N m 1 and 35 kg s 1 for stiffness and

13 Erickson, Weber, and Sharf / Contact Stiffness and Damping Estimation 53 Force (N) x 1 3 Displacement (m) Fig. 11. Experimental profiles of force and wall deflection with sinusoidal excitation. Stiffness (N/m) Damping (kg/s) Fig. 12. Experimental results of stiffness and damping estimation with sinusoidal excitation.

14 54 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / January 23 Force (N) Displacement (m) x Fig. 13. Experimental profiles for long contact maneuver of force and wall deflection with sinusoidal excitation. Stiffness (N/m) Damping (kg/s) Fig. 14. Experimental results for long contact maneuver of stiffness and damping estimation with sinusoidal excitation.

15 Erickson, Weber, and Sharf / Contact Stiffness and Damping Estimation 55 Stiffness (N/m) Damping (kg/s) Fig. 15. Experimental results of stiffness and damping estimation with PRBS excitation. Stiffness (N/m) Damping (kg/s) Fig. 16. Experimental results for long contact maneuver of stiffness and damping estimation with PRBS excitation.

16 56 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / January 23 Stiffness (N/m) Damping (kg/s) Fig. 17. Experimental results of stiffness and damping estimation without persistent excitation. damping, resulting in errors of 37% and 82%, respectively (Erickson 2). The substantial discrepancies from the actual parameter values can be explained by the fact that the impedance of the arm is not exactly represented by the target impedance. Also, finite sampled data and inaccuracies inherent in the estimation of the damping ratio limit the accuracy of the signal processing technique. Stiffness and damping estimation for the indirect adaptive and methods without persistent excitation is shown in Figure 17. The indirect adaptive controller resulted in stiffness estimates with an average error of 8.9% while the algorithm was more substantially amiss and demonstrated an average error of 19.2%. The cause for the discrepancy between the converged stiffness of the indirect adaptive method and the steady-state value predicted during mid-contact of the maneuver has not been uncovered. The results of the damping estimation were in line with our expectations. In the absence of persistent excitation, accurate damping estimation should not be possible with either algorithm. This was in fact observed with the indirect adaptive and methods which resulted in damping errors of 6% and 85%, respectively. However, an attempt at adaptation is visible during the initial and, in the case of, final transients, when the wall velocity is non-zero. Since the adaptation rate is gain-dependent, it cannot be concluded that the actual damping value will always be obtained during this limited time. 6. Conclusions The analysis and results presented in this paper are intended to compare contact parameter identification algorithms. Specifically, a signal processing method, indirect adaptive controller, MRAC controller and estimator were used to determine the stiffness and damping of the environment during robot constrained motion. The signal processing method represents an original contribution, while the remaining three schemes have been proposed by other authors (Seraji and Colbaugh 1997; Singh and Popa 1995; Love and Book 1995). However, substantial modifications were made to the original indirect adaptive controller, and a proof of parameter convergence was given. The signal processing method uses frequency-domain and time-domain information, and accordingly was implemented off-line. It has one substantial advantage: only force measurements are required to extract the desired information. The other three methods are time-domain algorithms and were implemented on-line. Their data requirements include the force, deflection and, with the exception of, velocity at the contact point. Accurate deflection or position measurements are difficult in practice, but many identification methods in the literature require it. In our future work, we will address this issue by including environment location in the parameters to be estimated.

17 Erickson, Weber, and Sharf / Contact Stiffness and Damping Estimation 57 The simulation results obtained with the four algorithms indicate that all can generate accurate estimates of both K e and B e in a short time. With the exception of the signal processing method, persistent excitation was found to be required for accurate estimation of both stiffness and damping. Without persistently exciting signals, stiffness estimation was still possible (in fact, it was more accurate), but damping estimation was not reliable. Three of the algorithms considered were tested experimentally by using a 3-DoF robotic arm contacting a flexible wall. The results of these tests indicate that the indirect adaptive and methods are capable of generating very good estimates for the wall stiffness and damping (in the presence of persistent excitation), within several periods of oscillation. Without persistent excitation, some damping estimation may occur during contact transients, but this depends on the maneuver and gain selection. The indirect adaptive solutions tended to be smoother, showing a more definite convergence. This, however, is likely to be due to the force-tracking feature of the controller, rather than an inherent property of the estimator. We also observe that the indirect adaptive controller was found to be intuitive and easily implemented, more so than the and MRAC schemes. Estimates calculated with the signal processing method were less accurate because of the discrepancy between real and target impedances of the manipulator and limited sampled data. Even then, we believe the method may still provide a viable option for practical applications, if the real impedance of the arm can be modeled. Our future work in the area of contact parameter identification will involve investigation of frequency-domain methods and evaluation of these and timedomain techniques for contacts involving complex geometries and stiff environments. References An, C., Atkinson, C., and Hollerbach, J Model-Based Control of a Robot Manipulator. Cambridge, MA: MIT Press. Anderson, B., Bitmead, R., Johnson Jr, C., Kokotovic, P., Kosut, R., Mareels, M., Praly, L., and Riedle, B Stability of Adaptive Systems: Passivity and Averaging Analysis. Cambridge, MA: MIT Press. Dorf, R., and Bishop, R Modern Control Systems. 7th edition. Reading, MA: Addison-Wesley. Erickson, D. 2. Contact stiffness and damping estimation for constrained robotics systems. Master s thesis, University of Victoria, Victoria, Canada. Gilardi, G., and Sharf, I. 22. Literature survey of contact dynamics modeling. Journal of Mechanism and Machine Theory, 37: Hogan, N Impedance control: An approach to manipulation: Parts i-iii. ASME Journal of Dynamic Systems, Measurement and Control, 17:1 24. Landau, I., Lozano, R., and M Saad, M Adaptive Control. London: Springer-Verlag. Ljung, L System Identification: Theory for the User. Englewood Cliffs, NJ: Prentice-Hall. Love, L., and Book, W Environment estimation for enhanced impedance control. In Proceedings of the IEEE International Conference on Robotics and Automation, pp Ma, O., Buhariwala, K., Roger, N., MacLean, J., and Carr, R MDSF a generic development and simulation facility for flexible, complex robotic systems. Robotica, 15: Seraji, H., and Colbaugh, R Force tracking in impedance control. International Journal of Robotics Research, 16(1): Singh, S., and Popa, D An analysis of some fundamental problems in adaptive control of force and impedance behavior: Theory and experiments. IEEE Transactions on Robotics and Automation, 11(6): Slotine, J.-J., and Li, W Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall. Van Vliet, J., Sharf, I., and Ma, O. 2. Experimental validation of contact dynamics simulation of constrained robotic tasks. International Journal of Robotics Research, 19(12): Weber, M., Ma, O., and Sharf, I. 22. Identification of contact dynamics model parameters from constrained robotic operations. Presented at DETC 2 ASME 22 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Montreal, Canada, September.