In this thesis, along the general direction of free space modeling using potential models, var-ious applications of potential models are investigated. The common idea of these algorithms is to use the repulsion exerted on an object, in forms of repulsive force and torque, from free space boundaries to achieve the best shape match between them. The object can be rigid or articulated, and the best match in shape is accomplished by adjusting object configuration, i.e., location and orientation, for minimum potential. Depending on the dimensions of free space, the degrees of freedom of the object, the extent of location change of the object (end-effecter), and the possible size change of the object, algorithms investigated in this thesis are summarized in Table 1.1.
The remainder of this thesis is organized as follows. In Chapter 2, the adopted potential field models presented in [13] [1] are reviewed. In 2-D environment, the free space are mod-elled by considering the potential due to both uniform and non-uniform source distributions on polygonal region borders of robots and obstacles. In 3-D environment, the free space are modelled by considering the potential due to uniform source distributions on surfaces
of robots and obstacles. The repulsion experienced by an object of finite size due to the potential gradient is considered. It is shown that the repulsion between polygonal object and obstacles, in forms of force and torque, can be derived in closed form.
In Chapter 3, applications of the Newton potential model [18] to path planning of a manipulator in 2-D work space are considered. The repulsive force and torque between the robot manipulator and obstacles are used to adjust the position and orientation of the former so as to keep it away from the latter. In Chapter 4, a potential-based path planning algorithm is proposed for 3-D manipulators. The proposed approach uses one or more guide planes (GPs) among obstacles in the free space as final or intermediate goals in the workspace for the articulated object to reach. These GPs provide the articulated object a general direction to move forward and also help to establish certain motion constraints for adjusting robot configuration for minimum potential during path planning. An extension of the path planning approach is proposed in Chapter 5 to derived a collision-free path for articulated robots with moving bases. In Chapter 6, a shape matching approach of 3-D object using the same 3-D potential model is presented which identifies the object shape with respect to a set of shape templates. The approach allows the object to grow in size inside a shape template and adjusts the object configuration according to the repulsive force and torque until a potential minimum is reached. Chapter 7 summarizes this thesis.
Chapter 2
Reviews of Adopted Potential Models
2.1 A Review of the 2-D Potential Model
2.1.1 Repulsion due to Charged Polygonal Region Borders
Polygonal description of obstacles is often used in path planning because of its simplicity. The repulsive force and torque between these polygonal regions can be derived by superposing the repulsion between pairs of border segments, each contains one line segment from the the moving object and the other from one of the obstacles. In general, each pair of the repelling line segments can have arbitrary configuration in the workspace as shown in Fig. 2.1(a).
To simplify the expressions of the repulsion between them in the following subsections, a coordinate system is chosen so that the obstacle line segment, AB, lies on the base line, as shown in Fig. 2.1(b). In the new coordinate system (uv-plane), the coordinates of the endpoints of AB are assumed to be (0, 0) and (d > 0, 0), respectively, and the line containing the object line segment , CD, can be represented as v = au + b, u1 ≤ u ≤ u2.
Figure 2.1: Coordinate transformation. (a) The original coordinate system (xy-plane). (b) The new coordinate system (uv-plane) after the transformation.
2.1.2 Integral equations for forces and torques
where ρ(u) is the charge density along AB.
For the total force on the object line segment CD, we have1 Fu = above integral equations can be formulated as
Fu =√ Fig. 2.1(b). The torque with respect to P , due to the repulsive force from AB on point Q is equal to
1For simplicity, only the u-component is considered for the rest of this thesis.
It is shown in [13] that with the above integral equations, the repulsive forces and torques between a rigid object and obstacles can be evaluated analytically for linear and quadratic charge distributions along their boundaries, as reviewed next. Thus, for the application of the proposed potential model in achieving collision avoidance in path planning, the optimal object configurations along a path can be found more efficiently.
2.1.3 Repulsion due to linear and quadratic charge distributions
Assume the charge density ρ(u) is equal to 1, u, or u2 for an obstacle line, and ρ(s) = 1, s, or s2 for an object line. Nine different combinations of charge distributions will need to be considered in evaluating the repulsion between the two line segments. For example, from (2.5), the repulsive force along the u-axis for these nine combinations can be obtained from
Fuij=Fuij(u2) − Fuij(u1) where i is equal to the order of the charge density of the obstacle line, and j is equal to that of the object line. It is shown in [13] that analytic expressions exist for all these integral equations. For example, we have
In general, ρ(u) and ρ(s) can be any quadratic functions, or
ρ(u) = α1u2+ β1u + γ1 (2.10)
ρ(s) = α2s2 + β2s + γ2 (2.11)
where coefficients α1, β1, γ1, α2, β2, and γ2 are some real numbers. The repulsive force along the u-axis can still be evaluated analytically as
Fu = α1α2Fu22+ α1β2Fu21+ α1γ2Fu20 +β1α2Fu12+ β1β2Fu11+ β1γ2Fu10
+γ1α2Fu02+ γ1β2Fu01+ γ1γ2Fu00 (2.12)
Similar results can be obtained for Fv and τP. Thus, for any ρ(u) and ρ(s), we can first eval-uate the coefficients of the charge density functions and then use the nine sets of expressions of the repulsion, as in (2.12), to evaluate the total repulsion.