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2.2. Inverse kinematics

Table 1 2.2. Inverse kinematics

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The schematic of the designed robotic CPR device is illustrated in Fig. 1. This mechanism has a mobile platform in the button of the robot, a fixed base at the top, and three limbs of identical RRPaR kinematic structure with the first R joint actuated by a rotary actuator. We use Pa and R stands for the notation of parallelogram and the revolute joint, respectively. Fig. 1. The designed asymmetrical spatial parallel robot.
The schematic of the designed robotic CPR device is illustrated in Fig. 1. This mechanism has a mobile platform in the button of the robot, a fixed base at the top, and three limbs of identical RRPaR kinematic structure with the first R joint actuated by a rotary actuator. We use Pa and R stands for the notation of parallelogram and the revolute joint, respectively. Fig. 1. The designed asymmetrical spatial parallel robot.
As shown in Fig. 2, it is assigned a fixed frame O{x, y, z} at the point O of the fixed base AA1A2A3. In the same way, a moving frame P {u, v, w} at the point P of the mobile platform AC|C2C3, with x, y axis parallels to u, v axis, respectively, and the z and w axes are perpendicular to the platform. Both the moving platform and fixed base are designed as an isosceles triangle described by the parameter of r = OA; and h = PC;, respectively, for i =/, 2, and 3. The actuated variable of the ith limb is q);. The connecting joints between the upper and the lower links denote by B;. In addition, a=A;B; and b=B;C; are the lengths of upper and lower links of each limb respectively. Fig. 2. Schematic representation of a 3-RRPaR parallel manipulator.
As shown in Fig. 2, it is assigned a fixed frame O{x, y, z} at the point O of the fixed base AA1A2A3. In the same way, a moving frame P {u, v, w} at the point P of the mobile platform AC|C2C3, with x, y axis parallels to u, v axis, respectively, and the z and w axes are perpendicular to the platform. Both the moving platform and fixed base are designed as an isosceles triangle described by the parameter of r = OA; and h = PC;, respectively, for i =/, 2, and 3. The actuated variable of the ith limb is q);. The connecting joints between the upper and the lower links denote by B;. In addition, a=A;B; and b=B;C; are the lengths of upper and lower links of each limb respectively. Fig. 2. Schematic representation of a 3-RRPaR parallel manipulator.
In this part of the article, the inverse kinematics of the proposed parallel manipulator is considered. The joint angles related with the ith limb are defined, as it is shown in Figure 3. where p is the point P of the mobile platform position vector. Coordinate system (xi, yi, z;) is attached to the base at point A; of the robot platform, such that the x;-axis is in line with the extended line of 04, the y,-axis is directed along the revolute joint axis at A;, and therefore the z-axis is parallel to the z-axis. The angle ¢;, is measured from the x-axis to the x;-axis and it is a constant constraint in manipulator design. Similar to the same method, qi is measured from the x;-axis to AB, qi is defined from the extended line of AB to the line defined by the intersection of the x;-z; and plane of the parallelogram, and finally, g3;is measured from the y; direction to =a
In this part of the article, the inverse kinematics of the proposed parallel manipulator is considered. The joint angles related with the ith limb are defined, as it is shown in Figure 3. where p is the point P of the mobile platform position vector. Coordinate system (xi, yi, z;) is attached to the base at point A; of the robot platform, such that the x;-axis is in line with the extended line of 04, the y,-axis is directed along the revolute joint axis at A;, and therefore the z-axis is parallel to the z-axis. The angle ¢;, is measured from the x-axis to the x;-axis and it is a constant constraint in manipulator design. Similar to the same method, qi is measured from the x;-axis to AB, qi is defined from the extended line of AB to the line defined by the intersection of the x;-z; and plane of the parallelogram, and finally, g3;is measured from the y; direction to =a
For inverse kinematic, the position vector p of the mobile platform is given and the problem is to find
For inverse kinematic, the position vector p of the mobile platform is given and the problem is to find
Where ¢ denotes the time variable, and x, y and z are in units of meters. According to the parameters of Table 1, the simulation results of the kinematic model are shown in Figs. 4, 5, and 6. These Figs are shown the joint angular displacements, velocities, and accelerations, respectively, which are derived via inverse kinematic closed-form solutions and inverse kinematics mode of MSC.Adams© software.
Where ¢ denotes the time variable, and x, y and z are in units of meters. According to the parameters of Table 1, the simulation results of the kinematic model are shown in Figs. 4, 5, and 6. These Figs are shown the joint angular displacements, velocities, and accelerations, respectively, which are derived via inverse kinematic closed-form solutions and inverse kinematics mode of MSC.Adams© software.
3. Jacobian of the manipulator Fig. 5. Verification results for angular velocities of (a) joint | (b) joint 2, and (c) joint 3.
3. Jacobian of the manipulator Fig. 5. Verification results for angular velocities of (a) joint | (b) joint 2, and (c) joint 3.
Fig. 7. Inverse kinematic singularities (a) Completely stretched, and (b) Completely contracted configuration. 4.2 Forward kinematic singularities o Forward kinematic singularities are associated with the forward Jacobian, J,, from Eq. (17), and arise: corresponds to the following condition:
Fig. 7. Inverse kinematic singularities (a) Completely stretched, and (b) Completely contracted configuration. 4.2 Forward kinematic singularities o Forward kinematic singularities are associated with the forward Jacobian, J,, from Eq. (17), and arise: corresponds to the following condition:
Fig. 8. A singular configuration in which all the lower links are in the plane of the moving plate. The second set of postures that satisfies Eq. (22) is found by the following condition
Fig. 8. A singular configuration in which all the lower links are in the plane of the moving plate. The second set of postures that satisfies Eq. (22) is found by the following condition
In this singular configuration, any two of the lower links are parallel to each other. Fig. 9 shows such ¢ singular configuration. Fig. 9. Singular configuration in which two or three lower links are parallel to each other. 4.3 Intermediate Jacobians and singularities — In this section, intermediate Jacobians find the relation between the velocity of the end-effector and the time rate of change of the length of a kinematic chain. This idea was introduced in (Lopez et al., 2006). To apply this idea for the modified Delta mechanism, the closed-loop OAiC; P S OAiBiC;P, which is obtained by ignoring the limbs a and b in the full kinematic chain OA;BiC;P, is reviewed. A loop-closure equation (1) for ith limb again can be rewritten as:
In this singular configuration, any two of the lower links are parallel to each other. Fig. 9 shows such ¢ singular configuration. Fig. 9. Singular configuration in which two or three lower links are parallel to each other. 4.3 Intermediate Jacobians and singularities — In this section, intermediate Jacobians find the relation between the velocity of the end-effector and the time rate of change of the length of a kinematic chain. This idea was introduced in (Lopez et al., 2006). To apply this idea for the modified Delta mechanism, the closed-loop OAiC; P S OAiBiC;P, which is obtained by ignoring the limbs a and b in the full kinematic chain OA;BiC;P, is reviewed. A loop-closure equation (1) for ith limb again can be rewritten as:
Fig. 10. Top view of the workspace of (a) original Delta, and (b) modified Delta robot. Fig. 10(a) and 10(b) show the top view of the workspace of the original Delta, and modified Delta mechanism respectively with architectural parameters which have been mentioned in Table 1. As it is seen in Fig. 10, compared with the original Delta mechanism, the obtained workspace has an asymmetrical character and its longitudinal direction is more extended than its lateral direction. This asymmetrical characteristic is affected by changing the position of three active joints relative to each other and it can be utilized in many applications with the special maneuver.
Fig. 10. Top view of the workspace of (a) original Delta, and (b) modified Delta robot. Fig. 10(a) and 10(b) show the top view of the workspace of the original Delta, and modified Delta mechanism respectively with architectural parameters which have been mentioned in Table 1. As it is seen in Fig. 10, compared with the original Delta mechanism, the obtained workspace has an asymmetrical character and its longitudinal direction is more extended than its lateral direction. This asymmetrical characteristic is affected by changing the position of three active joints relative to each other and it can be utilized in many applications with the special maneuver.
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Computer ScienceMechanism in BiologyPARALLEL MANIPULATOR
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