Optimal Excitation and Identification of the Dynamic Model of Robotic Systems with Compliant Actuators

Transcription

1 2015 IEEE International Conference on Robotics and Automation (ICRA) Washington State Convention Center Seattle, Washington, May 26-30, 2015 Optimal Excitation and Identification of the Dynamic Model of Robotic Systems with Compliant Actuators Jonas Vantilt 1, Erwin Aertbeliën 1, Friedl De Groote 1 and Joris De Schutter 1 Abstract An increasing number of robotic systems are using compliant actuators in which springs are placed in series with the actuator. The need for identification procedures tailored to these systems is consequently rising. When measurements of both the link side and the motor side of the spring are available the dynamic parameters can be identified independent of the spring model. The excitation of the identification procedure is optimized for the identification of the dynamical parameters to provide a rich data set by maximizing a reduced information matrix. Adopting this reduced matrix leads to a near-optimal excitation. A Fourier series is chosen as parametrization of the excitation, resulting in a periodic movement of the system under identification. This periodicity is exploited to develop an alternative weighting of the parameter estimation. This proposed weighting uses the motor torque variance at each time step of the trajectory, instead of one global torque variance. This identification procedure is demonstrated and validated on one leg of a lower limb exoskeleton. Only the dynamic model in the sagital plane (2D) is identified. I. INTRODUCTION Applications involving human-robot interaction are an emerging field in robotics. Assistive robotic devices like exoskeletons or prostheses are finding their way to the patient. In human-robot interaction the robot is not allowed to behave as a stiff, immovable object, due to safety considerations [1]. Therefore, a common feature of assistive robotics is the introduction of compliant actuators in the robot in order to increase the inherent compliance of the robot. Compliant actuators have some kind of flexible element between motor and link, like Series Elastic Actuators (SEA) or Variable Stiffness Actuators (VSA). Van der Kooij et al. [2] describe the advantages and disadvantages of compliant actuators and their use in exoskeletons. Van Ham et al. [3] provide an overview of compliant actuators. Another feature of assistive robotic devices is their adjustability to the user. For example, the link lengths of exoskeletons have to be tuned to each user. These adaptations change the dynamic model which thus has to be identified regularly, while taking into account motor and joint limits. An accurate dynamic model plays a crucial role in simulations and controller design of a robotic system. Some feedback control designs, like computed torque control, heavily rely on the dynamic model. Feedforward control also requires a system model in order to generate the inputs for accurate tracking. In summary, there is a growing need for automatic identification methods targeting these human-robot interaction applications. These methods should not depend on any manual tuning. 1 Department of Mechanical Engineering, KU Leuven, 3001 Heverlee, Belgium: firstname.lastname@kuleuven.be Identification techniques can be classified in several groups [4]. The objective of identification is the estimation of the parameters of a mathematical, dynamic model. Most of the work done on robot identification focuses on parameter estimation from dedicated experiments. In this case, the designed joint trajectories excite the robot and measurement data are collected. The design of the excitation is often overlooked, but is an important aspect of identification, since it influences the identifiability. The condition number of the regression matrix is a measure for the sensitivity of the result to perturbations on the regression matrix and measurements. If this sensitivity to disturbances becomes too high, the corresponding parameter estimation becomes unreliable. There are two methods to harvest a rich data set from the system. Either the identification method employs multiple experiments [5] or the method only performs one dedicated experiment to identify all parameters at once [6]. If the identification is composed of multiple experiments, parameters estimated in previous steps are used in successive experiments and identification steps. The drawback is the lower accuracy of the parameters, since the estimation errors influence the successive identification steps. Additionally, the identification becomes less automatic, since the configuration of the measurement setup changes between experiments and each experiment needs its own tuned excitation. On the other hand, when the identification only excites the robot in a single experiment, proper excitation design is harder. Therefore it is advised to resort to an optimization of this excitation. Swevers et al. [6] [7] advocate a way to optimize the excitation of rigid robots (i.e. without compliant actuators) by making use of Fourier series as parametrization of this excitation. This parametrization ensures a periodic trajectory with several benefits, see Section II-C. The identification of flexible joint robots is mostly split up in several steps with separate excitations. Albu-Schäffer et al. [8] design multiple dedicated identification steps for the flexibilities and friction terms. Wernholt et al. [9] first identify a rigid body model. Next, a frequency analysis is performed to identify the flexible parameters and the final step improves all the parameters. Gautier et al. also propose a three-step method to identify a flexible robotic system [10]. In this identification they make use of their DIDIM method [11], which only uses torque data to identify the system. It is basically a closed loop output error method which reduces the error between the measured and simulated torque. It involves a non-linear recursive optimization. Good starting values of the stiffness and inertia are necessary. Other work on flexible joint robot identification analyses the /15/$ IEEE 2117

2 Frequency Response Function (FRF) but the resulting models are only locally applicable as they linearize the system [12]. The contributions of this work are the following. An optimization technique for the excitation trajectory [6] is adjusted to flexible joint robots such that feasible, near-optimal excitation trajectories can be automatically generated. Due to the use of a reduced model for the excitation and due to the non-convexity of the optimization problem, the resulting trajectories are not globally optimal. In practice this does not pose a problem. Also an alternative weighting of the weighted least squares estimation is proposed. The excitation and identification are demonstrated on a bilateral lower limb exoskeleton with flexible joints (compliant element). Section II describes the methodology of the identification technique. Identification of an exoskeleton leg using the proposed method is covered in Section III. The practical implementation issues are also discussed in this section. Section IV discusses the result and Section V states the conclusion and future research. II. METHODOLOGY This section covers the proposed methodology for identification of robots with compliant actuators. This methodology is intended to identify both the dynamic model and the spring characteristics of the flexible joint system using only a single measurement campaign. The parameter estimation itself is composed of two steps using the data from the single experiment. In a first step the whole dynamic model, except for the spring equation, is identified. In a second step the previously identified dynamic model allows identifying the spring characteristic. The resulting spring parameters are thus dependent on the previously identified dynamic model. This identification requires the measurements of joint and motor angles (preferably after transmission) and torque or current measurements. A. Modeling and model reduction A first assumption arises in the modeling. The flexibilities of the links are negligible compared to the flexibilities in the joints, i.e. the links are modeled as rigid bodies. The modeling starts from the dynamic model for a robot with n flexible joints in joint space formulation [13]: H(q) q + S(q) T θ + C(q, q) q + C1 (q, θ) q + G(q) +τ fr,j ( q) = τ spring, S(q) q + B(q) θ + C 2 (q, q) q + τ fr,m ( θ) + τ spring = τ mot, with τ spring the spring torque, q is the n-vector of joint angles and θ is the n-vector of motor angles. H represents the n x n inertia matrix, S the n x n coupling matrix between joint and motor. The C i -matrices represent Coriolis and centrifugal terms. C(q, q) q represents the centrifugal and Coriolis terms as if the system was rigid. C 1 (q, θ) q and C 2 (q, q) q contain terms arising from the coupling S(q). G represents the gravity n-vector. The τ fr -terms represent the contribution of friction. The subscripts refer to the source (1) (2) of friction: either from the joint side j or from the motor side m. From here on, only viscous and coulomb friction are considered. The i th joint has the following joint friction: q i τ fr,j ( q)(i) = c v,j,i q i + τ coulomb,j,i tanh( ). (3) q off A hyperbolic tangent approximates the coulomb friction instead of a sign function in order to make this coulomb friction term differentiable. The q off 2 represents the minimum angular velocity at which the approximation error of the tangent hyperbolic function is smaller than 4%. This offset velocity should be chosen small. This will guarantee that the coulomb friction only deviates from a sign function at very low velocities. The identification of the dynamic model is independent of the stiffness model, because the spring torque is an internal torque of the system. When all state variables are measured, this torque is not required to model input-output behavior. The accuracy and complexity of the applied stiffness model will not influence the identification of the dynamic parameters. Consequently τ spring can be eliminated from the set of dynamic equations (1) and (2), leading to: H(q) q + S(q) T θ + C(q, q) q + C2 (q, q) q + τ fr,m ( θ)+ S(q) q + B(q) θ + C 1 (q, θ) q + G(q) + τ fr,j ( q) = τ mot. (4) The remainder of the paper discusses a reduced model with only planar motion and with the motor axes parallel to the joint axes. This is an approximate for the exoskeleton system in the experiments. In the planar case, the coupling inertia and the motor inertia become independent of q: S(q) = S, B(q) = B. Also, the Coriolis and centrifugal matrices C 1 and C 2 become zero matrices [13]. The resulting simplified equation can be written as: H(q) q + S T θ + S q + B θ + τfr,m ( θ) + C(q, q) q+ G(q) + τ fr,j ( q) = τ mot. To provide torque measurements for the parameter estimation the user relies on torque sensors on the motor if they are available. An alternative is to use current measurements to estimate the motor torques, using the equation: (5) τ current = k t I, with k t the torque constant. (6) As the motor is not overloaded, this linear expression to calculate motor torque is applicable. An overloading of the motor leads to a saturation of the motor flux and the torquecurrent curve starts to deviate from a linear relation. For the identification the efficiency of the transmission is also taken into account. Depending on the direction of energy flow, the transmission efficiency enters the torque equation as follows [7]: { τ current η if sign( τ mot = θ τ current ) > 0 τ current η if sign( θ τ current ) < 0. (7) Using a barycentric representation of the inertia parameters, equation (5) becomes linear in the parameters [14]. 2118

3 The simple viscous and coulomb model also leads to parameters, which enter the model linearly. The dynamic model (5) can thus be rewritten in a linear parametrized form: Φ(q, q, q, θ, θ) ψ = τ mot. (8) ψ represents the parameter r-vector which contains friction parameters, barycentric parameters and motor inertias. Φ is called the n x r regression matrix and is dependent on joint angles, velocities and accelerations and on motor velocities and accelerations. The motor torques are grouped into τ mot. B. Parameter estimation The parameter estimation involves the reduction of the residual of the error between model and the measurements. Different methods are available to minimize this residual error. In the Least Squares (LS) parameter estimation the influence of noise is neglected, which is a hard assumption. The Weighted Least Squares (WLS) only considers noise on the torque measurements (right hand side of (8)). In this case the regression matrix Φ is free of noise and becomes deterministic. The most common weighting for WLS [6] [15] [16] is to use the global torque variance of each actuator on the diagonal of the weighting matrix. If noise is present (and not negligible) on both the torque measurements and joint and motor angles one needs to resort to Maximum Likelihood Estimators (MLE) [6] [15]. In most robotic systems the noise on encoders is negligible compared to the noise on the torque measurements. Therefore the WLS estimator is in most cases the most appropriate estimator. In all these estimators the noise is considered Gaussian and uncorrelated. The WLS estimator of independent parameters is given: ψ wls = (F T Σ 1 F ) 1 F T Σ 1 b, (9) and in a more general case, using the peudo-inverse: = (Σ 0.5 F ) + Σ 0.5 b, (10) with F the grouped regression matrix, containing the regression matrices at each time step of the trajectory: Φ(q(1), q(1), q(1), θ(1), θ(1)) F =. (11) Φ(q(N), q(n), q(n), θ(n), θ(n)) and with b the torque measurements vector: [ T b = τ T (1)... τ (N)] T. (12) Numbers 1 to N indicate the index of the sampled measurements. The weighting matrix Σ contains the covariances of the measured actuator torques. The covariances of the actuator torques are all located at the diagonal of the weighting matrix, since the noise on the torques is assumed independent of the other actuators and independent over time. The uncertainty on the WLS ψ wls is described by the covariance matrix: COV wls = (F T Σ 1 F ) 1. (13) Before estimation, the matrix F is normalized by dividing its columns by their norm. The dynamic parameters are scaled accordingly. The condition number of the normalized regression matrix F is a measure for the sensitivity to disturbances on F and b. A high condition number indicates that one or more parameters are far more sensitive to disturbances and noise and thus will be badly identified. To verify that all parameters are independent a singular value decomposition is addressed. This SVD enables us to analyze the covariance COV wls and to detect possible linear dependencies between parameters. A proper construction and analysis of the regression matrix already gives much information about the validity of the results. C. Optimization of the excitation Most of the time the excitation is assumed to excite the robot system in such a way that a rich data set is obtained. With a naive excitation it is difficult to excite all frequencies and at the same time remain within physical feasible bounds. Hence, several trajectories have to be tested and tuned in order to get satisfactory results. In this paper, the excitation trajectory is optimized. The optimized trajectory improves the identification by ensuring a rich data set and keeping the condition number of F low. Second, it automatically generates feasible trajectories, no further tuning of these trajectories is required. Third, the optimization enables us to account for physically constraints like acceleration limits, velocity limits, range of motion limits, etc. For the optimization we have n motors to control and 2n free variables q and θ. One can only impose n independent trajectories to the systems. The relationship between q and θ is the spring characteristic. In order to optimize the excitation trajectory for the whole dynamic model, the θ angles should be transformed to joint angles q (or vice versa). This is done using the spring characteristics. However, in the excitation optimization step the dynamic parameters are unknown and thus the spring torque cannot be known in this excitation optimization step. Therefore, the proposed optimization is suboptimal, because it employs a reduced regression matrix Φ red in which the motor angles θ are eliminated: Φ red (q, q, q) ψ red = H(q) q + S q + C(q, q) q +G(q) + τ fr,j( q). (14) The reduced form is obtained from (5) by lumping the viscous and coulomb friction to the joint side (τ fr,j ) to remove θ from the reduced model. Second, the coupling S T θ and motor inertia B θ terms are eliminated. For removing the coupling and motor inertia terms S T θ + B θ, we assume that the parameters of these terms are known and can thus be added to the right hand side of (8). This is justified because the coupling and motor inertia matrices S and B only depend on the motor inertia parameter which can be found in motor data sheets. The resulting reduced regression matrix is now only dependent on q and can be optimized for these joint angles. The excitation trajectory will not be completely optimal for all parameters. However, low condition numbers 2119

4 for the regression matrices are obtained using these nearoptimal trajectories. The excitation trajectory to be optimized, is parameterized as a Fourier series. In discretized form the trajectory is: q i(k) = N i l=1 a i l ω f l sin(ω f lkt s) bi l ω f l cos(ω f lkt s) + q i0, (15) N i q i(k) = a i l cos(ω f lkt s) + b i l sin(ω f lkt s), (16) l=1 N i q i(k) = a i lω f l sin(ω f lkt s) + b i lω f l cos(ω f lkt s). (17) l=1 Where ω f is the fundamental pulsation of the Fourier series, which is the same for all joints. l indicates the order of the harmonic, k is the discrete time and T s is the sampling time. Furthermore, q i0 is the zero order harmonic or the average of the Fourier series. The index i indicates the number of the actuator. The symbols a i l and bi l are the coefficients of the sine and cosine harmonics of order l of actuator i. Due to the use of a Fourier series, the resulting trajectories are band limited, periodic trajectories. The advantages of this periodic robot excitation are: Time-domain averaging of the signals, which reduces the signal to noise ratio of the measured data. This is especially important for the motor current measurements used for the torque estimations since they are noisy. Calculation of the joint velocities and accelerations can be done in an analytic way, using the Fourier transform. Because the excitation is periodic, no leakage errors are introduced. Estimation of the noise characteristics of the measurements. Due to the periodicity of the signal the variances of the measurement noise can be calculated by exciting the robot system multiple periods. The Fourier series of each joint trajectory contains 2 N i +1 variables to be optimized: the coefficients a i l and b i l and q i0. The optimization criteria is the maximization of the reduced information matrix Fred T F red, without the weighting matrix Σ, as it is not known during the optimization of the excitation trajectory. As this is a matrix optimization a d- optimality criteria is adopted, minimizing the logarithm of the determinant: min a i l,bi l,qi0 log(det(f T redf red )), (18) with the reduced regression matrix as: Φ red (q(1), q(1), q(1)) F red =.. (19) Φ red (q(n), q(n), q(n)) This d-optimality [17] maximizes the determinant of the regression matrix. This equals to the maximization of a volume of the information of the system. This is also equal to the minimization of the volume of the variance ellipsoid of the parameters. torque [Nm] time [s] (a) Average motor torque (red) versus the motor torques of 20 excitation periods (blue), plotted on top of each other. velocity [rad/s] motor torque variance x2 [(Nm) 2 ] motor velocity motor torque variance x time [s] (b) Motor velocity versus motor torque variance. The velocity is shown in blue and is expressed in rad/s. The torque variance is the green curve and is given in (Nm) 2, the variance is multiplied by a factor of 2 for plotting purposes. The torque variance is high when the velocity is near zero. Fig. 1. The plots show the motor velocity, torque and torque variance during a periodic excitation of an exoskeleton (hip motor) under identification. Only half of the period is shown to improve the visibility. D. Trajectory-specific weighting of joint torques Fig. 1(a) shows the hip motor torque of the experiment performed in Section III. The blue curves are 20 periods plotted on top of each other. The red curve represents the averaged motor torque of these 20 periods. The blue curves indicate the spread of motor torques, calculated at each sample of the periodic motion. As can be seen, the motor torque variance is not equal over the whole trajectory. At certain points on the trajectory the variance is higher (e.g. at 3.7s) compared to the average. As Fig. 1(b) shows with the green curve, this higher variance occurs when the velocity (blue curve) crosses zero. The change of direction also results in a change of friction, which is a complex phenomenon, leading to a great variability of torque at these velocity reversal locations. It is thus more appropriate to use the torque variance at each sample of the trajectory instead of a 2120

5 global torque variance as is done in the literature [6] [15]. We can only calculate the variance at each sample of the trajectory due to the repetition of this periodic trajectory: σ 2 τ,i(k) = P (τ i (k + l N) ˆτ i (k)) 2 /(P 1) (20) l=1 with i the joint number, k the time index (k = 1... N), N the number of samples in the periodic trajectory, P the number of periods, l the current period and ˆτ(k) the average torque over the P periods at time index k. The green curve in Fig. 1(b) is a plot of the hip torque variance (20). The new proposed weighting of the WLS takes the torque variances at each sample of the trajectory as variances in the weighting matrix Σ. These variance enter the weighting matrix on its diagonal, since the actuators are independent. It is advised to select a low amount of harmonics, around ten. A higher amount not only leads to more coefficients to optimize, but it also increases the amount of passings of the zero velocity (velocity reversal). Because simple friction models do not capture friction effects at the velocity reversal (like stiction and hysteresis), these zero velocity passings should be avoided. This new weighting ensures that these zero passings are taken less into account in the parameter estimation. The motor torques have higher variances at these locations and thus these velocity reversals are weighted less in the parameter estimation. E. Spring characteristic When linear-in-the-parameter models are used the spring parameter estimation can be performed efficiently by LS or WLS. A simple linear stiffness model is not realistic as most compliant actuators have some form of spring stiffening. Therefore, we use a third order system to model the spring: τ spr = K 1 (θ q) + K 2 (θ q) 3 + D ( θ q). (21) The second order term is not present, since we assume a symmetric spring behavior under tensile and compressive load. The identification of the spring behavior has three parameters per joint to be identified. The spring equation can be written in parametrized form as follows: Φ spr (q, q, θ, θ) ψ spr = τ spr. (22) The spring torques can be calculated using the dynamic model identified in the first step of the identification process. The parameter estimation follows a similar approach as the estimation for the dynamic model. The WLS estimator of the spring parameters is given by: ψ spr,wls = (F T sprσ 1 sprf spr ) 1 F T sprσ 1 sprb spr, (23) with F spr the regression matrix similar in form as F in equation (11) and b spr the torque measurements vector, similar in form as equation (12). Fig. 2. Setup of the exoskeleton leg identification. The leg is suspended from the hip link. III. EXPERIMENTS AND RESULTS This section illustrates the identification method by applying it to an exoskeleton leg. The exoskeleton leg consists of three joints based on the Maccepa principle [18] and four links. The joints are equipped with encoders which allow measurements of the joint and motor angles. The motors are velocity controlled, but current measurements are available. The exoskeleton leg moves in the plane, which allows us to use the reduced planar dynamics (5). The leg is mounted on the thigh link as can be seen in Fig. 2. The transmission ratios and motor torque constants k t are known. A. Formulation of the dynamic model The first step is to set up the dynamic model. This planar model is derived using the Lagrangian energy equations. The equations can be rewritten in the form of equation (5), with the number of joints n equal to three. The equations obtained by the Lagrangian can also be written in parametrized form Φ(q, q, q, θ, θ) ψ. Alternatively, one can obtain the parametrized form from the Newton-Euler equations [19]. In this case not all the barycentric parameters can be identified because the exoskeleton is a planar system and because only joint torques are measured. However, not all the barycentric parameters are needed in the control either. Recall that the joint torques are derived using equations (6) and (7). Of the ten barycentric parameters for each moving link only three are needed in the dynamic model of the planar exoskeleton: m res,i L x,res,i, m L y,res,i and I zz,res,i. There are three moving links leading to a total of nine barycentric parameters for the exoskeleton leg. m res,i L x,res,i and m L y,res,i represent the barycentric moments (mass times length) and I zz,res,i is the resulting barycentric moment of inertia around 2121

6 the joint axis. An expression for barycentric parameters can be found in [14]. Due to the lumping of the friction effect to the joint side in the optimization (19), the friction effects are hard to separate again to motor and joint side in the parameter estimation. However this does not pose a big issue, since the lumped friction describes both friction effects. The friction model consists of coulomb and viscous friction (3). This gives two parameters per joint or a total of six parameters for the exoskeleton leg. The last parameters are the inertias of the motors which are present in both the coupling matrix S and the motor inertia matrix B. The three motor are identical and only one motor inertia parameter needs to be added. The total amount of model parameters that need to be identified becomes 16. B. Optimize the excitation trajectory In a second step the excitation trajectory is determined as the solution of an optimization. The trajectory is parametrized by a Fourier series. The base frequency is taken as 0.1Hz, rendering a period of 10s. Ten harmonics are taken (N i = 10), which result in 63 variables, since each joint has 2 N i + 1 (= 21) variables. The optimization uses the reduced regression matrix F red. The sample period is taken as 0.005s, resulting in 2000 discrete samples in the 10s period. The optimization can be written as: min a i l,bi l,qi0 log(det(f T redf red ) (24) subject to 0 q hip 2π 9 [rad] π 6 q knee 0 [rad] π 9 q ankle π 2 [rad] π q hip π [rad/s] π q knee π [rad/s] π q ankle π [rad/s] π q hip π [rad/s 2 ] 1.5π q knee 1.5π [rad/s 2 ] 2π q ankle 2π [rad/s 2 ], with the joint trajectories given as in equations (15) till (17) and the regression matrix given in (19). This constraint optimization enables us to take into account the physical limitations of the robot. To reduce gravitational torque at the hip joint, we add the following constraint: q hip + q knee 0.1rad. (25) This constraint ensures that when the hip joint is large the knee joint is small. This reduces the gravitational torque on the hip contributed by the shank and foot link. The formulation above is a non-convex optimization problem with many inequality constraints. It is solved in Matlab using fmincon with the active-set method, because of the large amount of inequality constraints of which only a small part is active any time. The optimization of angle [rad] All angles theta1 q1 theta2 q2 theta3 q time [s] Fig. 3. The averaged joint and motor angles are shown. The motor angles are solid lines and the joint angles are represented by dashed lines. The blue curves are from the hip, the green curves are from the knee and the red curves are from the ankle. The black lines represents the optimized excitation trajectories. The motor angles θ are tracking these optimized trajectories. the excitation is non-convex. Finding a global optimum is therefore not guaranteed. When the stopping criterion is reached after several 100 iterations, the objective function log(det(f T red F red) equals The condition number κ of the regression matrix with the optimized trajectory is The resulting trajectory is shown in Fig. 3 by the thin black lines. C. Execution of the excitation trajectories The third step involves the execution of the optimized trajectories. In the previous step the excitation is optimized for the joint angles. These optimized trajectories are now applied to the motor velocity with a PD controller with feedforward. The compliance in between motor and joint causes the joint angles to deviate from the optimized trajectory as is seen in Fig. 3. The dashed lines are the motor angles which follow the black optimized trajectories. The joint curves (in full lines) follow a different trajectory from the optimized ones. Ideally we would like to apply the optimized trajectory directly to the joints. This would require a tracking controller using the dynamic model which we first have to identify. However, applying the optimized trajectory to the motor angles still results in a low condition number κ and a rich measurement data set, see Subsection III-D. The periodic trajectory is imposed for several minutes before the data are recorded to make sure that the exoskeleton moves in regime. Twenty periods of data are recorded. D. Data processing and parameter estimation Time averaging is applied to the periodic trajectories. Afterwards a frequency domain filtering is applied to the joint and motor angles. Fitting the joint and motor angles with a Fourier series enables us to calculate the derivatives of the angles analytically. This prevents the noise build-up 2122

7 inherent to taking finite differences. Due to the flexibilities between motor and link, the joint angles contain more frequencies than the motor angles. This is taken into account by selecting more frequencies in the data processing than present in the motor angles. 17 harmonics are taken for the Fourier series of the data processing in contrast with the 10 harmonics N i in the optimization. This number of harmonics is tuned: taking too much harmonics leads to over fitting and much noise, while taking too few harmonics throws away the higher frequency information of the excitation. The regression matrix F is constructed using the filtered joint and motor data. The motor currents are also averaged and converted to motor torques using (6) and (7). The motor torques are collected in the torque measurement vector. The condition number κ of the regression matrix is which approximates the condition number from the optimized trajectory (slightly better). The natural logarithm of the determinant of F T F equals When the weighting Σ is applied the logarithm of the determinant of F T Σ 1 F equals 241. As discussed in Subsection II-D and Fig. 1(a) and 1(b), the weighting matrix Σ is constructed using the variance of motor torques at sample of the periodic movement, instead of calculating one global variance for the whole trajectory as is done in the literature [6] [15] [16]. This leads to a more appropriate weighting of the parameter estimation as it takes parts of the trajectory with higher uncertainty less into account. For example, velocity reversals are weighted less, see Subsection II-D. After the data processing and construction of the weighting matrix, the weighted least squares parameter estimation is performed according to (9). The results of the parameter estimation are listed in table I. The table also contains the standard deviation of each parameter. The variance of the parameters is calculated with (13). IV. VALIDATION AND DISCUSSION The obtained parameter values are validated in several ways. Table II gives an overview of the investigated performance criteria. The first validation is to compare the identified parameters with the CAD data. The results of the identification correspond well with the CAD models, see Table I. All the identified parameters are larger than the parameters of the CAD data. This is expected, since the CAD model doesn t take into account the controller boards of the motors, the parts for mounting these controller boards, the shielding parts and the cabling of sensors and motor. Parameters of the commercial braces were manually approximated. The motor inertia from the data sheet is in accordance with the identified one, which is another indication of the validity of the parameter estimation. Also note that the friction parameters have higher variances and thus have larger error bands. A second validation uses the identified parameters to estimate the motor torque of a different trajectory. A typical gait trajectory is chosen for the validation. This trajectory is applied to the joint angles by a computed torque and feedforward controller which both use the dynamic model. TABLE I COMPARISON OF THE PARAMETER ESTIMATION AND CAD DATA parameter CAD data value spread σ m 1 L res,1,x (kgm) e 4 m 1 L res,1,y (kgm) e 4 I zz,res,1 (kgm 2 ) e 4 m 2 L res,2,x (kgm) e 4 m 2 L res,2,x (kgm) e 4 I zz,res,2 (kgm 2 ) e 4 m 3 L res,3,x (kgm) e 4 m 3 L res,3,x (kgm) e 4 I zz,res,3 (kgm 2 ) e 4 I mot (kgm 2 ) e 4 c v,1 (Nm/s) e 3 c v,2 (Nm/s) e 3 c v,3 (Nm/s) e 3 τ coulomb,1 (Nm) e 3 τ coulomb,2 (Nm) e 3 τ coulomb,3 (Nm) e 3 K 1,hip (Nm/rad) K 2,hip (Nm/rad 3 ) D hip (Nms/rad) K 1,knee (Nm/rad) K 2,knee (Nm/rad 3 ) D knee (Nms/rad) TABLE II OVERVIEW OF PERFORMANCE CRITERIA performance criteria opt. traj. true traj. old Σ new Σ log(det(f T F )) ( ) N.A. N.A. κ(f ) ( ) N.A. N.A. RMS τmot (Nm) N.A. N.A The error between the modeled and actual torque is expressed as a RMS value and is equal to Nm. This RMS value is smaller than the RMS value obtained when parameters, resulting from a classical weighting, are used (i.e N m). This is an indication of the improvement of the proposed weighting. The torques reach 15N m during the gait trajectory. The condition number and determinant of the regression matrix of the optimized trajectory and the true measurement data are compared. The condition number of the regression matrix F of the measurements is slightly better than the one from the optimization: compared to This indicates that the near-optimality of the optimization is not a real issue and it also shows that applying these optimal joint trajectories to the motor does not affect the optimality of the excitation too much. The identified spring characteristics (see table I) deviate 2123

8 from the values of the provider. These compliant actuators are not commercially available ones, but they are custom made for the exoskeleton at hand. As a consequence variability in the production is expected, causing a variability in the resulting spring characteristics. The data obtained by the provider originates from experiments on a single actuator. Therefore it is hard to asses the validity of the spring characteristic. The spring characteristic of the ankle joint is not presented as it is hard to identify as the spring is hardly excited (due to low inertia of the foot), see Fig. 3. V. CONCLUSION AND FUTURE WORK An excitation optimization procedure for identification of robots with compliant actuators is presented in this paper. The excitation is optimized towards the dynamical parameters. The identification requires measurements of motor and joint angles and current measurements of the motor. Measuring both the joint and motor angles allows to make the identification of the dynamic parameters independent of the spring characteristic. The proposed method optimizes the excitation trajectories for robots with compliant actuators. These trajectories are parametrized as Fourier series and the periodicity offers several benefits. Although the global optimization is not guaranteed (non-convex), the resulting excitation provides good data sets. Finally, an alternative weighting of the Weighted Least Squares is proposed by exploiting the periodicity. The presented techniques are applied to an bilateral lower limb exoskeleton with compliant actuators. Future work involves validating the identification for a more general setup e.g. a setup where the motor axes are not parallel to link axes and setups which are no longer planar. However, their is no methodological problem to apply this work to 3D systems. Also the influence of the proposed new weighting, using the torque variance at each sample of the periodic trajectory, will be further investigated. The optimization of the excitation now mainly looks at the dynamical parameters. The spring is also excited well by the applied trajectory. However taking the spring parameter identification into the optimization of the excitation would improve the spring identification. ACKNOWLEDGMENT The authors gratefully acknowledge the financial support by the Flemish agency for Innovation by Science and Technology through a project grant (MIRAD, IWT-SBO ) and a Ph.D. grant (IWT-SBO ). [4] J. Wu, J. Wang, and Z. You, An overview of dynamic parameter identification of robots, Robotics and Computer-Integrated Manufacturing, vol. 26, pp , [5] J. Ghan, R. Steger, and H. Kazerooni, Control and system identification for the berkeley lower extremity exoskeleton (bleex), Advanced Robotics, vol. 20, no. 9, pp , [6] J. Swevers, C. Ganseman, D. Bilgin, J. 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