Page 306 - Handbook of Biomechatronics

298 Andres F. Ruiz-Olaya et al.

system—Cartesian, {x,y,z}, cylindrical,{r,θ,z}, or spherical, {ρ,φ,θ} can
be used to fully determine robot position. The selection of a particular coor-
dinate system depends on the kinematic structure of the robot (Pons, 2008).
In robotics, the preference is to describe the position and orientation in a
more compact form based on the translation and rotation of the coordinate
frame. A rotation matrix, R, is a transformation matrix that, when multiplied
by a vector, has the effect of changing the direction of the vector but not its
magnitude. A rotation is an orthonormal transformation in which the oppo-
site rotation is represented by the transposed version of the original matrix,
T
R 1 ¼ R (Pons, 2008). The analysis of kinematics of robots is usually based
on homogeneous transformation matrices. In jointed, multilink mechanisms
like robots, the relative motion of links around a joint can be simply
described by homogeneous transformation matrices. Finding the form of
the forward kinematic problem for a robot can be approached by the
Denavit-Hartenberg (D-H) convention. The D-H convention establishes
an algorithm for assigning a set of coordinate systems which are related
through translation and rotation transformations. The transformation
between successive coordinate systems takes into account the particular
kinematics of robot joints, as shown in Eq. (1) (the general form of the
transformation matrix between two consecutive coordinate systems)
(Pons, 2008).

2 3
cosθ i cosα i sinθ i sinα i sinθ i α i cosθ i
T i ¼ 6 sinθ i cosα i θ i sinα i cosθ i α i sinθ i 7 (1)
7
6
i 1 4 0 sinα i cosα i d i 5
0 0 0 1
When considering multibody, jointed mechanisms like wearable robots,
dynamics deals with the analysis of movement in a configuration, and work-
ing space as a function of internal forces (e.g., torque at each joint actuator)
and external forces (e.g., force interactions with the environment) (Pons,
2008). Two instances of the relationship between force and movement
can be identified: the forward dynamics problem and the inverse dynamics
problem. The forward dynamic model expresses the evolution of joint and
working coordinates as a function of the force and torque involved. The
Inverse dynamics model describes forces and torques as a function of the
evolution of joint coordinates in time (Pons, 2008).
In robotics, Newtonian and Lagrangian mechanics are used to derive the
dynamic model of a robot. The Newton-Euler formulation is based on a