A well-established model of the moment, , generated by applying stimulation, u(t), to a muscle acting about a single joint, , is
where h(u(t), t) is a Hammerstein structure incorporating a static non-linearity, , representing the isometric recruitment curve, cascaded with linear activation dynamics, . The multiplicative term captures the effect of joint angle and angular velocity on the force generated. When multiple joints are actuated by multiple muscles and/or tendons which may each span any subset of joints, then the general expression for the total moment generated about the ith joint is
Here is the moment arm of the jth muscle with respect to the ith joint, where E is the associated excursion (displacement) [8]. If each muscle length is primarily dependent on a single joint angle, the form can be taken, leading to the simplified structure
It is also possible to include the neuromuscular reflex in the form of an additional dynamic function placed in series with the muscle model. However it is neglected here since ES produces negligible effect on the reflex loop when applied on a macroscopic scale as in the transcutaneous case considered in [9, 10]. It is also worth noting that recent works have shown that Hill-Huxley models [11–13] may be at least as accurate as a Hammerstein structure in representing the activation dynamics [14]. The drawback that their complexity undermines application to control has been countered by the proposal of a Hammerstein-Wiener structure [15], but as yet Hill-Huxley models have not been shown to extend to non-isometric conditions, and have not been used in controller derivation.

As stated, the human arm is often supported by a mechanical device during ES assisted task practice in order to reduce fatigue and provide additional assistance.A general dynamic model of the support structure which assumes rigid links is
where is a vector of q joint angles, is a vector of externally applied force, and and are inertial and Coriolis matrices respectively. In addition, is the system Jacobian, and and are friction and gravitational vectors respectively. Finally, vector comprises the moments produced by the assistive action of the support mechanism. This may be passive, via springs or counter-balances, or active, as in the case of a robotic structure supplying active torque to assist, or even resist, the intended movement.

A popular form of support is an exoskeletal structure which enables assistance to be applied about individual joints. An example is the commercial ArmeoSpring (Hocoma AG) which provides adjustable force against gravity via two springs. Each joint is aligned in either the horizontal or vertical plane, as shown in Fig. 2.1a, with measured joint variables . The patient’s arm is rigidly strapped to the exoskeleton support with lengths , relating the shoulder joint to a fixed base frame.

Hence for the ArmeoSpring and are 5-by-5 inertial and Corelis matrices, and moments produced through gravity compensation provided by each spring yield the form . Figure 2.1c shows the axes corresponding to anthropomorphic joints.

Another common structure is the end-effector type where support is only supplied at a single attachment point. An example is the SaeboMAS (Saebo, Charlotte, USA) shown in Fig. 2.2. Here the support takes the form .

It is now assumed that within the necessary joint ranges there exists a unique bijective transformation between coordinate sets, given by , which allows the mechanical support and human arm models to be combined. This explicitly holds for exoskeletal passive or robotic structures (where ), and can be extended to end-effector robot devices developed for rehabilitation. The Lagrangian equation in one variable can be expressed in terms of the other through application of the chain rule, and the results added to produce the combined model
where
with and .

Now let each be realized using continuous-time state-space model matrices , , (state, input and output respectively), with corresponding states . The system (2.6) can then be expressed over time interval in the following state-space form
where , and the ith row of is given by

We next develop procedures to identify parameters in composite model (2.6) that can be used in a clinical setting. We first assume it is possible to manipulate each joint individually while measuring and recording the resulting joint angle and applied force signals. This is clearly not possible for all joints in the wrist and hand, and so alternative identification approaches for these structures are presented in Chap. 8.