Section 5.2 described the inverse position kinematic equations of the proposed micromanipulator. This section discusses the inverse velocity kinematics. If the mobile platform velocities are known the joint velocities can be computed using the inverse Jacobian matrix:
, (5.27) where is a 3 x 1 joint velocity vector, is a 6 x 1 mobile platform velocity vector consisting of the linear and angular velocity components and, is a 3 x 6 inverse Jacobian matrix of the micromanipulator.
This section derives first a general form of the inverse Jacobian which consists of six column vectors. Then two 3 x 3 inverse Jacobian matrices are derived based on the first and the second generation inverse kinematic models. The first generation inverse kinematic model assumes only translational motions and therefore, the first generation inverse Jacobian uses the first, second and third column of the general inverse Jacobian.
In the second generation kinematic model, the micromanipulator is assumed to have one translational and two angular motion components. Therefore, the inverse Jacobian which is based on this assumption includes the third, fourth and fifth column of the general inverse Jacobian. The two inverse Jacobians will be derived in Section 5.3.2 and Section 5.3.3.
5.3.1 General Form of the Inverse Jacobian Matrix
The Jacobian matrix can be derived by differentiating the link equations with time. The link equations are:
, (5.28)
∆zm
q· = J–1x·
q· x· = p·m θ· T
J–1
Bw
i = Bpi'+Bpm–Bbi
Chapter 5, Kinematics 78
where is the rotated upper mounting point in the base frame, is the position of the centre of the mobile platform, and is the position of the lower mounting point in the base frame.
The link vector can be presented in the form:
, (5.29) where is a unit vector in the direction of and is the magnitude (length) of the link vector.
Now substituting Equation (5.29) into Equation (5.28) and differentiating yields
. (5.30) Since the position of the lower mounting point does not change, .
The velocity of the upper mounting point can be represented as a cross product of an angular velocity vector of the mobile platform and a upper mounting vector as in Equation (5.31):
, (5.31)
where is the angular velocity vector of the mobile platform, and is the rotated position of upper mounting point of ith link.
Substituting Equation (5.31) into Equation (5.30) and then taking a scalar product with the unit vector on both sides of the equation, yields
, (5.32) since is a unit vector, and 1.
The inverse velocity kinematics can now be written in the form
, (5.33)
where is the link velocity vector, is the mobile platform velocity with linear and angular velocity components and the inverse Jacobian is as given in
1.
Chapter 5, Kinematics 79 Equation (5.34):
. (5.34)
The unit vectors are obtained as
. (5.35)
By substituting the simplified transformation matrix given in Equation (5.19) to Equation (5.10), which determines the link vectors, we obtain for the unit vectors:
, (5.36)
where r is the radius of the mobile circle, si and ci represent and with , and are rotations about the xB and yB axes, respectively, , and are translations along the xB, yB and zB axes, respectively, and represents the length of the link.
Since the cross product columns are not needed in the first generation inverse Jacobian, a general form of the columns is not derived here. They are presented for the second generation inverse Jacobian in Section 5.3.3.
5.3.2 First Generation Inverse Jacobian
Since the first generation model assumes that the mobile platform does not possess rotational degrees of freedom, the last three columns of the general inverse Jacobian matrix can be ignored.
The first generation inverse Jacobian is thus a 3 x 3 three matrix that consist of the first three columns, the unit vectors , of the general inverse Jacobian derived in Section 5.3.1:
Chapter 5, Kinematics 80
. (5.37)
Since the rotations do not take place, . Therefore, we have for the unit vectors:
. (5.38)
The inverse velocity kinematics is now
, (5.39)
where the inverse Jacobian is
. (5.40)
The first generation inverse Jacobian is derived from the first generation inverse position kinematic equations, the inverse Jacobian can also be derived by differentiating the inverse position kinematic equations given in Equation (5.14):
. (5.41)
where , . By differentiating Equation (5.41), we obtain
, (5.42)
Chapter 5, Kinematics 81 Equation (5.42) can be written in the matrix form as follows:
. (5.43)
As can be seen, the inverse Jacobian matrix is the same in Equation (5.43) as in Equation (5.40). The applicability of the matrix to position feedback control will be demonstrated in Chapter 7.
5.3.3 Second Generation Inverse Jacobian
The second generation inverse Jacobian is based on the second generation inverse kinematic model. In the second generation model, the mobile platform possesses two rotational and one translational degrees of freedom. Since the translations along the xB and yB axes are assumed to be zero, we obtain the following unit vectors, see Equation (5.36):
. (5.44)
Correspondingly, the cross-product columns are:
(5.45)
Chapter 5, Kinematics 82
The general form of the inverse Jacobian contains six
column vectors. However, only the third ( ), the forth ( ) and the fifth ( ) column are needed in this model. Using Equation (5.44) and Equation (5.45) we obtain:
(5.47)
(5.48)
(5.49)
By taking into account the values of the upper and lower mounting points given in Equation (5.6) and Equation (5.7), respectively, the columns of the inverse Jacobian are:
(5.50)
Chapter 5, Kinematics 83
(5.52)
Since the model is based on the assumption that the manipulator has one translational (along the z axis) and two rotational (about the x and y axes) degrees of freedom, the link velocities can be computed as follows:
. (5.53)
The inverse Jacobian derived in this section will be used as a part of the vision-system-based position feedback control scheme to be discussed in Chapter 7. Its feasibility as a static nonlinear decoupling element used after three linear single-input / single-output (SISO) controllers will be experimentally shown in Chapter 7.