Soutenance mémoire
Passive Rehabilitation Therapy
KINEMATICS AND DYNAMICS
In this chapter, we present the kinematic and dynamic modeling of the ETS-MARSE. The first section of this chapter describes the details of the kinematic modeling. Modified DenavitHartenberg (DH) notations were used to develop the kinematic model. The mid section of the chapter briefly explains the iterative Newton-Euler method which was used to develop the dynamic model of the ETS-MARSE. The chapter ends with a brief discussion on Jacobians, which map the joint space velocity with the Cartesian velocity.
Kinematics
To rehabilitate and ease human upper limb movement, the ETS-MARSE was modeled based on the anatomy and biomechanics of the human upper limb. Modified DH conventions were used in developing the kinematic model. The procedure of coordinate frame assignment (link frame attachment) and the definition of DH parameters are briefly summarized in the next subsection.
Coordinate Frame Assignment Procedure
There are different ways to assign coordinate frames to the manipulator links. For the ETSMARSE we have followed the Denavit-Hartenberg method. (Craig, 2005; Denavit and Hartenberg, 1955). The steps are as follows (Hartenberg and Denavit, 1964): • assume each joint is 1DoF revolute joint; • identify and locate the axes of rotation; • label the joint axes ܼ • locate the origin of each link-frame (Oi) where the common perpendicular line between the successive joint axes (i.e., ܼିଵ and ܼ) intersects. If the joint axes are not parallel, locate the link-frame origin at the point of intersection between the axes; • locate the Xi axis (at link frame origin Oi) as pointing along the common normal line between the axes ܼିଵ and ܼ. If the joint axes intersect, establish Xi in a direction normal to the plane containing both axes (ܼିଵ and ܼ); • establish the Yi axis through the origin Oi to complete a right-hand coordinate system.
Definition of D-H Parameters
A link of a robot can be described by four parameters (two parameters for describing the link itself and other two for describing the link’s relation to a neighboring link) if we assign the co-ordinate frames as described above (Denavit and Hartenberg, 1955). These parameters are known as Denavit-Hartenberg (DH) parameters. The definitions of the DH parameters are given below (Hartenberg and Denavit, 1964): 1, 2, and 3 together constitute the shoulder joint, where joint 1 corresponds to horizontal flexion/extension, joint 2 represents vertical flexion/extension, and joint 3 corresponds to internal/external rotation of the shoulder joint. The elbow joint is located at a distance de (length of humerus) apart from the shoulder joint. Note that joint 4 represents the flexion/extension of the elbow joint and joint 5 corresponds to pronation/supination of the forearm. As depicted in Figure 3.2, joints 6 and 7 intersect at the wrist joint, at a distance dw (length of radius) from the elbow joint, where joint 6 corresponds to radial/ulnar deviation, and joint 7 to flexion/extension. The modified DH parameters corresponding to the placement of the link frames (in Figure 3.2) are summarized in Table 3.1. These DH parameters are used to get the homogeneous transfer matrix, which represents the positions and orientations of the reference frame with respect to the fixed reference frame. It is assumed that the fixed reference frame {0} is
located at distance ds apart from the first reference frame {1}.
Inverse Kinematics
The inverse kinematics solution for a manipulator is computationally costly compared to direct kinematics. Due to the nonlinear nature of the equations to solve, it is often hard to find a closed form solution; sometimes multiple solutions may also exist (Siciliano, Sciavicco and Villani, 2009). Moreover, an inverse kinematics problem for a redundant manipulator is much more complex since it gives infinite solutions. We know that, for a manipulator having a square Jacobian, joint velocities can be found from the following relation (Craig, 2005) Cartesian velocity vector. Therefore, inverse kinematic solutions can be obtained easily bysimply integrating the joint velocities. The ETS-MARSE is a redundant manipulator; therefore it is not possible to find closed form solutions. Moreover, its Jacobian is not square, therefore we are not able to directly use Equation (3.6) to find joint positions. As an alternative approach, the inverse kinematic solution of the ETS-MARSE was obtained by using the pseudo inverse of Jacobian matrix ܬ)ߠ) (Siciliano, Sciavicco and Villani, 2009). For a redundant manipulator, the Equation (3.6) can be reformulated as (Siciliano, Sciavicco and Villani, 2009): where ܬற(ߠ (is the pseudo inverse generalized, and can be expressed .
Singularity Analysis
The ETS-MARSE arm will be in a singular configuration when it is straight down by the side (i.e., a singularity will occur when the axes of rotation of joint-1 (Z1), and joint-3 (Z3) , and/or joint-5 (Z5), and/or joint-7 (Z7) are aligned with each other). Note that the joint-space based control algorithms (that includes both linear and nonlinear control techniques e.g., PID control, computed torque control, sliding mode control) do not require a Jacobian matrix or inversion of a Jacobian matrix, therefore singularity is not a big issue in this case. On the other hand, Cartesian based control approaches, which are often used to maneuver the manipulator in a straight line motion, require Jacobian or inverse Jacobian matrices; therefore for this type of control the singularity must be properly dealt with. Interestingly to replicate these type of trajectories as a rehabilitative exercises e.g., to follow a square trajectory over the surface of a table, joints 2, 4 and 6 are usually far away from the singular configuration of the ETS-MARSE model. Note that anatomically rotation of joint-6 is limited to +20° to -25°. Moreover, as a safety measure when using Cartesian based control, a singularity could be easily avoided by limiting the position of joint-2, and joint-4 to more than 10° .
Dynamics
The studies of dynamics discuss the manipulator motions and the forces and torque that cause them. Among the various methods found in literature the iterative Newton-Euler formulation and the Lagrangian formulation are widely used to develop the dynamic model of a manipulator. Note that for a 6DoFs manipulator the Newton-Euler approach is 100 times (computationally) more efficient compared to the Lagrangian approach (Craig, 2005). Therefore, we have used the iterative Newton-Euler method (Luh, Walker and Paul, 1980) to develop the dynamic model of the ETS-MARSE. A brief overview of this method is given below.
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Table des matières
INTRODUCTION
CHAPTER 1 LITERATURE REVIEW
1.1 End-effector based Rehabilitative Devices (State of the Arts)
1.2 Exoskeleton type Rehabilitative Devices (State of the Arts)
1.3 Limitations of Existing Rehabilitative Devices and Robotic Exoskeletons
1.4 Research Objectives and Hypothesis
1.5 Passive Rehabilitation Therapy
1.6 Contribution
CHAPTER 2 MOTION ASSISTED ROBOTIC-EXOSKELETON FOR SUPERIOR EXTREMITY (ETS-MARSE)
2.1 General Design Considerations
2.2 Design Consideration for ETS-MARSE
2.3 Development of ETS- MARSE
2.4 Hardware implementation of ETS-MARSE
2.4.1 CAD Modeling
2.4.2 Simulation
2.4.3 Design
2.4.4 Fabrication
CHAPTER 3 KINEMATICS AND DYNAMICS
3.1 Kinematics
3.1.1 Coordinate Frame Assignment Procedure
3.1.2 Definition of D-H Parameters
3.2 Inverse Kinematics
3.3 Singularity Analysis
3.4 Dynamics
Iterative Newton-Euler Formulation
3.5 Jacobians
CHAPTER 4 CONTROL AND SIMULATION
4.1 PID Control
Simulation with PID
4.2 Compliance Control with Gravity Compensation
4.3 Computed Torque Control (CTC)
Simulation with CTC (Rahman et al., 2011f)
4.4 Modified Sliding Mode with Exponential Reaching Law (mSMERL)
4.4.1 Simulated results with SMC (Rahman et al., 2010c):
4.4.2 Simulated results with conventional SMERL
4.5 Cartesian Trajectory Tracking with Joint based Control
CHAPTER 5 EXPERIMENTS AND RESULTS
5.1 Experimental Setup and Control Implementation
5.2 Passive Rehabilitation Using Pre-determined Exercises
5.2.1 Experimental Results with PID Control (Rahman et al., 2011d; 2012c)
5.2.2 Experimental Results with Compliance Control (Rahman et al., 2012c)
5.2.3 Experimental Results with Computed Torque Control (Rahman et al., 2011c)
5.2.4 Evaluation of mSMERL Regard to Trajectory Tracking
5.2.5 Trajectory Tracking Performance Evaluation of PID, CTC, and mSMERL
5.3 Cartesian Trajectory Tracking (Rahman et al., 2012a)
5.4 Passive Rehab Therapy Using master Exoskeleton Arm (Rahman et al., 2011h)
5.4.1 Experimental Results with PID Control
5.4.2 Experimental Results with CTC
5.5 Discussion
CONCLUSION
RECOMMENADTIONS
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