2.3 Parameterization of manipulators via twists Using the product of exponentials formula the kinematics of a manipulator is completely characterized by the twist coordinates for each of the joints. We now consider some issues related to parameterizing robot motion using twist Choice of base frame and reference configuration In the examples above we chose the base frame for the manipulator to be at the base of the robot other choice of the base frame are possible and can sometimes lead to simplified calculations. One natural choice is to place the base frame coincident with the tool frame in the reference configuration. That is we choose a base frame which is fixed relative to the base of the robot and which lines up with the tool frame when θ=0. This simplifies calculations since g st (0)=I with this choice of base frame and hence A further degree of freedom in specifying the manipulator kinematics is the choice of the reference configuration for the manipulator. Recall that the reference configuration was the configuration corresponding to setting all of the joint variable to 0. By adding or subtracting a fixed offset from each joint variable. We can define any configuration of the manipulator as the reference configuration. The twist coordinates for the individual joint of a manipulator depend on the choice of reference configuration(as well as base frame) and so the reference configuration is usually chosen such that the kinematic analysis is as simple as possible. For example a common choice is to define the reference configuration such that points on the twist axes for the joint have a simple form as in all of the example above. Consider the scara manipulator with base frame coincident with the tool frame at θ=0 as shown in figure3.5. the twist are now calculated with respect to the new base frame: And similar calculation yield Expanding the product of exponentials formula gives Note that g st (0) = I which is consistent with the fact that the base and tool frame are coincident at θ =0. Compare this formula with the kinematics map derived in example3.1 Relationship with denavit-hartenberg parameters Given a base frame S and tool frame T the coordinates of the twist corresponding to each joint of a robot manipulator provide a complete parameterization of the kinematics of the manipulator. An alternative parameterization which is the de facto standard in robotics is the use of denavit-hartenberg parameters [25 ]. In the section we discuss the relationships between these two parameterization and their relative merits Denavit-hartenberg parameters are obtained by applying a set of rules which specify the position and orientation of frames L i attached to each link of the robot and then constructing the homogeneous transformations between frames denoted . By convention we identify the base frame s with L 0 . The kinematics of the mechanism can be written as just as in equation(3.1). Each of the transformations gl i-1, l i has the form Where the flour scalars α i ,ai,di and ɸ i are the parameters for the i th link. For revolute joints, ɸ i corresponds to the joint variable θ i . while for prismatic joint, d i corresponds to the joint variable θ i . Denavit-hartenberg parameters are available for standard industrial robots and are used by most commercial robot simulation and programming systems It may seem somewhat surprising that only four parameters are needed to specify the relative link displacements since the twist for each joint have six independent parameters. This is achieved by cleverly choosing The link frames so that certain cancellations occur. In fact it is possible to give physical interpretations to the various parameters based on relationships between adjacent link frames. An excellent discussion can be found in spong and vidyasagar [111] There is not a simple one to one mapping between the twist coordinates for the joint of a robot manipulation and the denavithartenberg parameters. This is because the twist coordinates for each joint are specified with respect to a single base frame and hence do not directly represent the relative motion of each link with respect to the previous link. To see this, let be the twist for the i th link relative to the previous link frame. Then gl i- 1l i is given by And the forward kinematics map becomes This is evidently not the same as the product of exponential formula though it bears some resemblance to it The relationship between the twist and the pairs and can be determined using the adjoint mapping. We first rewrite equation(3.8) as We can simplify this equation by making use of the relationship to obtain It follows immediately that ξ i = Ad gl 0l i− 1 ( 0) ξ i− 1 ,i . (3.10) This formula verifies that the twist ᶓ i is the joint twist for the i th joint in its reference configuration and written relative to the base coordinate frame Given the Denavit-hartenberg parameters for a manipulator the corresponding twists ξ i can be determined by first parameterizing g i-1,i using exponential coordinates as in equation(3.7), and then applying equation(3.10). However in almost all instances it is substantially easier to construct the joint twists ᶓ i directly by writing down the direction of the joint axes and in the case of revolute joints choosing a convenient point on each axis. Indeed one of the most attractive features of the product of exponential formula is its usage of only two coordinate frames the base frame S and the tool frame T. tis property combined with the geometric significance of the twists ξ i ,make the product of exponentials representation a superior alternative to the use of denavithartenberg parameters 2.4 Manipulator workspace The workspace of a manipulator is defined as the set of all end effector configurations which can be reach by some choice of joint angles. If q is the configuration space of a manipulator and g st :Q->SE(3) is the forward kinematics map then the workspace W is defined as the set W={ g st (θ): θ ϵ Q =(p,R) } ⊂ SE(3) (3.11) The workspace is used when planning a task for the manipulation to execute all desired motion of the manipulation must remain within the workspace. We refer to this notion of workspace as the complete workspace of a manipulator Characterizing the workspace as a subset of SE(3) is often somewhat difficult to interpret. Instead one can consider the set of positions(in R 3 ) which can be reached by some choice of joint angles. This set is called the reachable workspace and is defined as W R = {p(θ) : θ ∈Q}⊂R3 , (3.12) Where p(θ):Q -> R 3 is the position component of the forward kinematics map g st . The reachable workspace is the volume of R 3 which can be reached at some orientation Since the reachable workspace does not consider ability to arbitrarily orient the end effector for some task it is not a useful measure of the range of a manipulator. The dexterous workspace of a manipulator is the volume of space which can be reached by the manipulator with arbitrary orientation: WD = {p ∈R3 : ∀R ∈SO(3), ∃θ with g st (θ) = (p,R) }⊂R3 . (3.13) The dexterous workspace is useful in the context of task planning since it allows the orientation of the end effector to be ignored when positioning objects in the dextrous workspace For a general robot manipulator the dextrous workspace can be very difficult to calculate. A common feature of industrial manipulator is to Figure 3.6 workspace calculation for a planar three link robot (a). The construction of the workspace is illustrated in (b). the reachable workspace is shown in (c) and the dextrous workspace is shown in (d) Place a spherical wrist at the end of the manipulator as in the elbow manipulator given in example 3.2. Recall that a spherical wrist consists of three orthogonal revolute axes which intersect at a point. If the end effector frame is placed at the origin of the wrist exes then the spherical wrist can be used to achieve any orientation at a given end effector position. Hence for a manipulator with a spherical wrist the dextrous workspace is equal to the reachable workspace,W D =W R . Furthermore the complete workspace for the end effector satisfies W =W R ×SO(3) . this analysis only holds when the end effector frame is placed at the center of the spherical wrist if an offset is present the analysis becomes more complex Example 3.4 workspace for a planar three link robot Consider the planar manipulator shown in figure 3.6a. let g=(x,y, )ɸ Represent the position and orientation of the end effector. The forward kinematics of the mechanism can be derived using the product of exponentials formula, but are more easily derived using plane geometry: x = l1 cos θ1 + l2 cos ( θ1 + θ2 ) + l3 cos ( θ1 + θ2 + θ3 ) y = l1 s in θ1 + l2 s in(θ1 + θ2 ) + l3 s in(θ1 + θ2 + θ3 ) (3.14) φ = θ1 + θ2 + θ3 . We take l 1 >l 2 >l 3 , and assume l 1 >l 2 +l 3 The reachable workspace is calculated by ignoring the orientation of the end effector. To generate it, we first take and as fixed. The set of reachable points becomes a circle of radius l3 formed by sweeping angles to get an annulus with inner radius l2-l3 and outer radius l2+l3 centered at the end of the first link. Finally, we generate the reachable workspace by sweeping the annulus through all values of , to give the reachable workspace. The final construction is shown in figure3.6c. wr s an annulus with inner radius l 1 -l 2 - l 3 and outer radius l 1 +l 2 +l 3 . The dextrous workspace for this manipulator is somewhat subtle. Although the manipulator has the planar equivalent of a spherical wrist, the end effector frame is not aligned with the center of the wrist. This reduces the size of the dextrous workspace by 2l 3 on the inner and outer edges, as shown in figure 3.6d.