Technical noteA method for measuring endpoint stiffness during multi-joint arm movements
Introduction
Endpoint stiffness of the arm, resulting from the resistance of muscles to a small displacement of the hand, is an important mechanical property of the musculoskeletal system, characterizing resistance to disturbances encountered in the environment (Hogan, 1985). When the hand is perturbed slightly, it tends to return to its original position. Stiffness can be computed as the ratio of restoring force to displacement amplitude.
Using this technique, the features of endpoint stiffness during posture have been investigated extensively (Mussa-Ivaldi et al., 1985; Tsuji et al., 1995; Gomi and Osu, 1998). However, measuring stiffness during multi-joint movements remains a technical challenge. A rigid, powerful mechanical interface with computer-controlled dynamics is also needed to control hand position during movement (Gomi and Kawato, 1997).
Most previous estimates of stiffness during movement have been realized by means of force perturbations. Bennett and others (Bennett et al., 1992; Lacquaniti et al., 1993) have used stochastic force disturbances and measured the resulting change in hand position. This method is relatively straightforward to implement, but results in stiffening of the joints due to muscle co-contraction (Milner, 1993) and can reduce or even abolish the stretch reflex (Stein and Kearney, 1995). For these reasons, Bennett (1993) later employed a displacement relative to the mean undisturbed movement for determining joint stiffness during elbow movements. Although the mechanical interface for displacing the hand must be precisely controlled, only a few trials are needed and data analysis is straightforward (divide force by displacement amplitude).
Force impulses have been used to estimate stiffness during multi-joint movements (Gomi and Kawato, 1997). However, the method suffers from the limitation that a perturbation of the same amplitude, applied at different points in the trajectory or in different directions, will displace the hand by different amounts. This is because limb stiffness depends on joint angles, angular velocity and perturbation direction (Mussa-Ivaldi et al., 1985; Gomi and Kawato, 1997; Bennett, 1993). Since the stiffness depends on the displacement amplitude (Shadmehr et al., 1993), bias may result. Furthermore, damping and stiffness must be identified together, so many trials are needed with perturbations in multiple directions.
Consequently, we decided to adapt Bennett's method to study the stiffness of two-joint arm movements. This required consideration of nonlinear limb dynamics, and accurate two-dimensional prediction of where the trajectory would have gone, had there been no perturbation, in order to achieve a constant displacement. For one-joint movements, the mean of past trajectories is an acceptable estimate of the unperturbed trajectory, whereas this is generally not true for multi-joint movements because they are less constrained.
This paper presents the key features of our technique for estimating stiffness during movement: an accurate prediction algorithm for multidimensional trajectories and the implementation of a servo-controlled perturbation during movement. The method is tested in simulations and its efficiency for measuring stiffness without disrupting motion is demonstrated in experiments conducted with two subjects.
Section snippets
Methods
Let be the two-dimensional vector with the shoulder and elbow as first and second coordinates, and the corresponding two-dimensional Cartesian hand position vector. Let be the (endpoint) force required to move the arm along the trajectory , and let be the (endpoint) force developed by the muscles activated by to realize this movement. The components of correspond to all the muscles involved in the movement. If there is no interaction
Results
The simulation showed that the prediction error is small (<1 cm after 200 ms). The prediction error is sometimes positive, sometimes negative, and distributed symmetrically around the unperturbed velocity (Fig. 1). The mean prediction error is close to 0, and the mean force is close to the force corresponding to perfect trajectory prediction. This indicates that the prediction algorithm and the stiffness estimation are unbiased and accurate.
Tests with a perturbation of amplitude 0 showed that
Discussion
The method presented in this paper requires fewer trials to accurately estimate stiffness during movement than methods which have been used previously, including methods based on stochastic perturbations (Bennett et al., 1992; Lacquaniti et al., 1993). Force perturbations, such as those used by Gomi and Kawato (1997), are simpler to implement than our position perturbation. However, both stiffness and damping must be estimated, making the identification more difficult, since damping estimates
Acknowledgments
The experiments were performed at ATR. We thank H. Gomi for having set up the testbed, and the reviewers for their valuable remarks. This research was supported by ERATO/JST, Japan, JISTEC, Japan, and by grants from the Swiss National Science Foundation and the Natural Sciences and Engineering Research Council of Canada.
References (15)
Torques generated at the human elbow joint in response to constant position errors imposed during voluntary movements
Experimental Brain Research
(1993)- et al.
Time-varying stiffness of human elbow joint during cyclic voluntary movement
Experimental Brain Research
(1992) - Burdet, E., Osu, R., 1999. Development of a new method for identifying muscle stiffness during human arm movements....
- Burdet, E., Osu, R., Franklin, D., Milner, T.E., Kawato, M., 1999. Measuring stiffness during arm movements in various...
- et al.
Human arm stiffness and equilibrium-point trajectory during multijoint movement
Biological Cybernetics
(1997) - et al.
Task-dependent viscoelasticity of human multijoint arm and its spatial characteristics for interaction with environments
Journal of Neuroscience
(1998) The mechanics of multi-joint posture and movement control
Biological Cybernetics
(1985)
Cited by (145)
Hand-impedance measurements with robots during laparoscopy training
2022, Robotics and Autonomous SystemsCitation Excerpt :Based on such modelling, measuring human hand-impedance or arm joint impedances implies estimation of the mass, spring, and damping parameters within the aforementioned LTI model. Applying small impulse type force or position perturbations from a grip point and analysing the resulting response behaviour of the hand, such as interaction force and displacement from the equilibrium posture, is extensively revisited methodology; for instance see [15,21] for the force and [22–24] for the position disturbances. To apply perturbations and measure the hand position, admittance controlled robotic manipulators have been used in our previous studies [15,25] within the aforementioned techniques without considering the overall system’s stability.
Impedance Learning for Human-Guided Robots in Contact with Unknown Environments
2023, IEEE Transactions on RoboticsHuman Modeling in Physical Human-Robot Interaction: A Brief Survey
2023, IEEE Robotics and Automation LettersAfter a Decade of Teleimpedance: A Survey
2023, IEEE Transactions on Human-Machine Systems