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with every object represented as a sphere. The Cost function considers smoothness of the path, and the torque of the actuators, to ensure that the robot is not overstretched.

40.9.5 OPTIMIZATION : SENSOR BASED

An interesting approach to path planning was suggested by B.Espiau and R.Boulic [1986]. In the attempt to find a path planning method for a 10 degree of freedom manipulator, they came up with an unusual approach to optimization. With proximity or contact sensors mounted on every link of the manipulator, the path was navigated. To do this the sensor data would be used alter the penalty functions to represent detected obstacles encountered by the manipulator. The cost function was based on velocity and dynamic functions. This gave a dynamic approach to finding paths in which the path was found by the local optimization of trajectories. For this particular method a large number of sensors are required, and unfortunately the authors did not provide statistics about the performance of the method.

40.9.6 OPTIMIZATION : ENERGY

In a method suggested by M.Vukobratovic and M.Kircanski [1982] the energy of the system was proposed as an excellent method for path planning. Using the techniques of optimization for energy the best path is found. This is done by calculating the dynamics at certain points in space, and then using the dynamic programming technique (of operations research) to find the best path. This technique runs a few minutes on a PDP11/70. The end effector is ignored in this method as negligible. This method is only intended to smooth paths, so that the stresses on the load and manipulator are decreased. The information provided on this method was not very complete.

40.10 APPENDIX B - SPATIAL PLANNING

Spatial Planning is best described as making maps of space, then using the direct relationships between those objects in space to find paths. These methods cover a variety of techniques, but essentially their primary funtion is to determine the spatial relations between the object and the obstacle and avoid collisions. These techniques in general have not produced the best paths, but they produce paths quickly. These methods are also best used with 2D problems.

40.10.1 SPATIAL PLANNING : SWEPT VOLUME

Lozano-Perez and Wesley (1979) discuss a generate and test strategy used for path planning. The technique will begin with a straight line from start to goal, regardless of collisions. Then a repetitive cycle of analysing a path (by detecting collisions on the path with Swept Volumes) and then using the information to generate an improved path.

This method may be more formally described with a set of steps. The first step is to check the validity of the proposed path. The path validity is found by considering volume swept out as the object moves along the path. If a collision is not detected the method will be halted. In the case of collision, the information about the collision will be used for correction of path. This

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information used may include details about the collision, like shape of intersections of volumes, the object causing collision, depth of penetration, and the nearest free point.

The difficulties of this solution become obvious when some of the intricacies of the problem are considered. Models of complex surfaces can contain a very large number of simple surfaces. Calculating the intersections of these numerous simple surfaces can be a very difficult task. A second problem is how we may determine a global optimum when only local information about obstacles at collisions is made available. With the local information about collisions being used in path correction, radical different options are ignored. These two problems could result in an expensive search of the space of possible paths with a very large upper bound on the worst case path length.

Figure B.1 Swept Volume Path Planning

G		G
S	S	Swept
		Volume
Initial problem	First Iteration
G		G
S	S
Second Iteration	Third and last Iteration
40.10.2 SPATIAL PLANNING : OPTIMIZATION

Lozano-Perez and Wesley [1979] describe their work (see Cartesian Configuration Space) as being an improvement over the Optimization Technique of Spatial Planning. The basic concept

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they describe, is to explicitly compute the collision constraints on the position of a moving object relative to other obstacles. The trajectory picked is the shortest path which will satisfy all of the path constraints.

With objects modelled as convex polyhedra, vertices of the moving object may move between the obstacle surface planes and collide. This condition is easy to detect, because if a vertices is outside any plane of an obstacle, there is no collision. One possible simplification is to use a circle to represent the objects geometry, and just maintain a radial clearance from all objects. It should also be noted that the circular geometry is not sensitive to rotation. This was the path planning technique used in a mobile vehicular robot called SHAKEY by N.J.Nilsson [1969].

Figure B.2 Spatial Planning - Optimization

40.10.3 SPATIAL PLANNING : GENERALIZED CONES

Generalized cones [R.A.Brooks, 1983] are a faster approach than the Cartesian Configuration Space method . These cones are achieved through a special representation of the environment. The surfaces of convex polygons are used to determine conically bounded pathways, for the path of the object being moved. The method of determining free pathways (or Freeways) is based on the use of cone shaped spaces. The cones fit snugly between objects and have spines that run along the centre of the cones. These spines are the paths that the object may travel along. This makes the method inherently 2D and thus has not been implemented in 3D as of yet, but it has spawned a method which is successful in 3D by Brooks[1983]. To determine which spines to follow, the author uses the A* search technique to explore the various paths along the spines. This leads to problems in cluttered spaces where certain possible paths may be overlooked by the generalized cones. This method chooses a path with considerable clearance of objects.

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Figure B.3 Problem Represented with Generalized Cones

cone

spine

Figure B.4 Problem Represented with Generalized Cones

Goal

Start

As can be seen the rotation with this technique is very restricted, and the object is typically oriented with the spine.

40.10.4 SPATIAL PLANNING : FREEWAYS

A follow up to R.A.Brooks [1983] research into the use of Generalized cones for the representation of Free Space, R.A. Brooks [1983] developed a method of path planning for manipulators with 5 or 6 d.o.f. motion. This method is able to solve the pick and place problem in under a minute on an MIT Lisp machine, by approximating the robot as a 4 d.o.f. manipulator. His method is based on the assumption that the world is represented as supported, and suspended,