Short communicationThe concept of mobility in single- and double handed manipulation
Introduction
The concept of end-point mobility in a robotic or human manipulator was introduced by Hogan in three seminal papers on impedance control (Hogan, 1985a, Hogan, 1985b, Hogan, 1985c). The mobility represents the inertial behavior of the manipulator, and is the inverse of the inertia tensor for the end-point. A controller (human or automatic) may make use of this concept to control the manipulator such that the end-point shows larger resistance to disturbances in some directions. Hogan wrote
The physical meaning of the end-point mobility tensor is that if the system is at rest (zero velocity) then a force vector applied to the end-point causes an acceleration vector (not necessarily co-linear with the applied force) which is obtained by premultiplying the force vector by the mobility tensor.
It should be noted that a force acting on a moving manipulator will experience additional resistance due to velocity-dependent terms in the equations of motion. Think about the resistance to changes in orientation of the axis of a spinning wheel. Hogan׳s papers on impedance control deal with the problem of controlling a manipulator with significant dynamic interaction with the environment (the manipulated object). Humans perform such manipulation tasks with astonishing performance considering the significant signal delays and variability inherent in the neuro-muscular system. Hogan thus postulated (Hogan, 1984, Hogan, 1985b) that humans obtain good performance by not only making the end-point follow a desired path, but by also modulating the inertia and stiffness of the end-point, i.e. by impedance control. The possibility of modulating the inertia (and mobility), stiffness and damping of the end-point/end-link is a feature of redundant manipulators. Modulation of stiffness and damping is made possible by co-activation of antagonists.
A considerable number of papers have used the concepts of impedance and mobility in the study of motor control of human reaching tasks. See, for instance Sabes et al. (1998), Hamilton and Wolpert(2002), Lametti et al. (2007), Selen et al. (2009), and Cos et al. (2011). The experimental condition has typically involved a model of the arm as a planar three-link mechanism. We are currently using the concept of mobility to gain insight into the motor control of fast and accurate bi-manual tasks, using the golf swing as the specific case. This case differs from single-limb reaching tasks in the important aspect that the two arms collaborate. Hogan derived the expression for the mobility based on the Jacobian of an open-chain, single arm manipulator, but did not consider the case of two-arm manipulation. Zatsiorsky (2002) also discusses the concept of end-effector mobility, but only for open kinematic chains. There is thus a need to extend the theory of mobility to double-handed grasping and manipulation.
The purpose of this short communication is to present the method of calculating the mobility both for the end-point and for the whole end-segment and for both single- and double handed manipulation in a unified way so that this interesting concept may be more accessible to researchers in biomechanics and motor control.
Section snippets
Kinematics and kinetics
The kinematics and kinetics of the human body are approximated by a mechanical linkage, i.e. a chain of rigid bodies connected by joints. The mechanism has a number of degrees of freedom (DoF), n, represented by a vector of generalized coordinates .
The human body interacts physically with the surroundings. We are here primarily concerned with manipulation, and hence focus primarily on the contact between the hands, or implements held by the hands, and the surroundings. Importantly,
Mobility
The generalized manipulator inertia, , of a mechanical linkage is a symmetric and positive semi-definite matrix which depends, in general, on the configuration q of the linkage. Its inverse is the mobility matrix for the complete linkage, and denoted Y(q). Generalized momentum can now be defined aswith the inverse relationshipThe kinetic energy becomes
We next look at the mobility of a single point or link of the manipulator. This could be the end-point or
Example
The concept is illustrated with data from a professional golf player swinging at full speed with a wedge club. We show how the gripping of the club with two hands influences the mobility of the end-point, which was taken to be a point in the middle between the two hands.
Data were recorded using a Polhemus Liberty electromagnetic tracking system (Polhemus Inc., Colchester, VT, USA), sampling at 120 Hz. Seven sensors were attached at the lumbo-sacral joint, between the shoulders at the level of
Conclusion
The concept of mobility may reveal strategies of the motor control system in manipulation tasks. Redundant mechanisms such as the human arm can be configured in many different ways while performing the same task. The concept of mobility quantifies the degree to which a certain configuration promotes movement in certain directions, and resists (unwanted) movements in other directions.
This paper provides a general presentation of the method for both single-handed and double-handed manipulation.
Conflict of interest statement
None of the authors have any commercial or other interest which are in conflict with the integrity of this work.
Acknowledgments
The authors would like to thank Professor Emeritus Torsten Söderström (Uppsala University) and the reviewers for insightful discussion of the manuscript. The study was funded by the Swedish National Centre for Research in Sports (CIF) (Grants P2011-0191 and FO2012-0060).
References (14)
Adjustments to Zatsiorsky–Seluyanov׳s segment inertia parameters
J. Biomech.
(1996)- et al.
The influence of predicted arm biomechanics on decision making
J. Neurophysiol.
(2011) - et al.
Tracking the motion of hidden segments using kinematic constraints and Kalman filtering
J. Biomech. Eng.
(2008) - et al.
Using an extended Kalman filter for rigid body pose estimation
J. Biomech. Eng.
(2005) - et al.
Controlling the statistics of action: obstacle avoidance
J. Neurophysiol.
(2002) Adaptive control of mechanical impedance by coactivation of antagonist muscles
IEEE Trans. Autom. Control
(1984)Impedance controlan approach to manipulation: Part i theory
J. Dyn. Syst. Meas. Control
(1985)