Robot Modeling for Physical Rehabilitation MATHEMATICAL MODELING There are many design difficulties when develop- ing a proper mechanical system for moving an injured human limb, using either an industrial robot, an exoskeleton or a cable-based structure. Most of them are related to limb movement, which is imposed by its anatomy. This is because it is not so easy to locate the position of the rotational center of the human joint, in general the joint is modeled as a spherical joint or as a hinge joint, and located inside the body (Gopura & Kiguchi, 2009). Therefore, the first step to define the me- chanical system is related to the necessary motion to be performed, section 2. In therapeutic methods for movement rehabili- tation, the joint is moved very slowly by therapist or by an external-powered device. In order to avoid disuse atrophy, therapeutic device must have multiple degrees of freedom to allow the neces- sary motion of the limb. Any type of mechanical system used for rehabilitation must offer excellent control of movements and interaction of forces, allowing the adequate rehabilitation therapy. Industrial robots, generally, are controlled using point-by-point method in which interest lies in the position of its end effector. Then the end effector, a splint, must be well attached to the human limb, allowing the limb to perform the imposed robot motion. This kind of fixing is not always appropriate because the limb is already injured and can compromise patient comfort. Position control strategies can be used when there is a null or weak interaction between the robot and the patient. Therefore, one can use compliant devices, which preserve the causal relationship between patient effort and the resulting limb movement. According to Rosati et al. (2009), they are easier to design, cheaper, and safer compared to exo- skeletons, but they do not allow control of single human joints and cannot measure human displace- ments, velocities, torques, etc. Another alternative that has been considered as the most adequate technique for physiotherapy, applied to control the interaction of the robot and the patient, is the impedance control that was first proposed by Hogan (1985), which aims to specify the relationship between position and force. In the following subsections, the usual math- ematical modeling of mechanical structures ap- plied to human limb rehabilitation are presented. As many articles analyze serial and closed loop articulated mechanisms, including workspace and trajectory optimization, control methodologies and so on, one presents considerations about them. Although impedance control has been around since 1985 (Hogan, 1985), its application for in- teraction between robots or exoskeletons should be analyzed. Therefore impedance control is presented. The description of a cable-based par- allel structure is also presented in order to show its applicability in the rehabilitation process with results from the experimental setup. Modeling Mechanical Systems for Motion Rehabilitation In general, industrial robots are based on serial architectures. These robotic structures have been studied for years and their kinematic models can be described with matrix methods, using the ho- mogeneous transformation matrix from Denavit- Hartenberg coordinate systems, for example, or the method of successive screw displacements. The dynamic model can be obtained from Newton- Euler laws or by the Lagrangian formulation, which are well depicted in robotic books, such as those of Tsai (1999) and Angeles (1997). One alternative to promote the rehabilitation movements consists of using an exoskeleton based on articulated mechanism (or linkages), which has been studied since antiquity and is still the subject of several studies (Ceccarelli, 2007). Its use for exoskeleton can be justified due to its capability of developing motions, from linear to more complex, while transferring force and/or torque. 163