# Design, Analysis, and Applications of Mobile Manipulators

Tao Song (Shanghai University, China), Feng Feng Xi (Ryerson University, Canada) and Shuai Guo (Shanghai University, China)
DOI: 10.4018/978-1-5225-5276-5.ch002
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## Abstract

Presented in this chapter is a method for design and analysis of a mobile manipulator. The wrench induced by the movement of the robot arm will cause system tip-over or slip. In tip-over analysis, three cases are considered. The first case deals with the effect of the link weights and tip payload on the horizontal position of the CG. The second case deals with the effect of the joint speeds through the coupling terms including centrifugal forces and gyroscopic moments. The third case deals with the effect of the joint accelerations through the inertia forces and moments. In slip analysis, the first case considers the reaction force in relation to the stand-off distance between system and work-piece. The second and third cases investigate the effects of the joint speeds and accelerations. Then, the mobile platform is optimized to have maximum tip-over stability which optimizes the placement of the robot arm and accessory on the mobile platform. The effectiveness of the proposed method is demonstrated.
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## Wrench Modeling

### Manipulator Kinematics

Figure 1.

Manipulator kinematic modeling with static and motion parts

Figure 1 shows a kinematic model of the manipulator under this study. The method presented in (Xi, 2009; Lin, Xi, Mohamed, & Tu, 2010) is used here for kinematic modeling. This method formulates the manipulator kinematics through two parts. The first is a static part to represent the initial configuration of each link and the second is a motion part to represent the movement of each joint. For the static part, a set of initial configuration set-up (ICSU) are defined including a static rotation matrix and a static body vector , for each link, that is

(1)
(2) where , , are three rotation matrices about X, Y, and Z axis of local frame relative to local frame , and , , are the three unit vectors of local frame attached to the th joint. is a vector representing the th link at the initial configuration. In this chapter, a bold vector is expressed with respect to the base frame, and a bold vector with an apostrophe is expressed with respect to a local frame.

The total translation and total rotation of the th link is expressed by including the motion part as

(3)
(4) where and are the motional rotation matrix and motional body vector, respectively. In practice, corresponds to a rotational joint driven by a rotary motor and corresponds to a prismatic joint driven by a linear motor. In case of revolution joint, is equal to , and is the rotation angle of the th joint.

To this end, the end-effector’s position and orientation , can be expressed with respect to the base frame of the mobile platform as

(5)
(6) where represents the rotation matrix of link with respect to the base frame, which is multiplied sequentially by a number of . In eqn. (5), subscript indicates the tip of the last link where the end-effector is located, and the end-effector’s orientation coincides with the last link, expressed by .

Taking the time derivative of eqn. (5) and (6) leads to the following forward recursive velocity equations

(7)
(8) where is the velocity vector at the th joint and is the angular velocity vector of the th link. Note that the following holds

(9)
(10)

Furthermore, taking the time derivative of eqn. (7) and (8) results in the following forward recursive acceleration equations

(11)
(12) where is the acceleration vector at the th joint and is the angular acceleration vector of the th link.

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