Planar Kinematics-Modern Robotics-Lecture Handouts, Lecture notes of Introduction to Robotics

Download Planar Kinematics-Modern Robotics-Lecture Handouts and more Lecture notes Introduction to Robotics in PDF only on Docsity! Chapter 4 Planar Kinematics Kinematics is Geometry of Motion. It is one of the most fundamental disciplines in robotics, providing tools for describing the structure and behavior of robot mechanisms. In this chapter, we will discuss how the motion of a robot mechanism is described, how it responds to actuator movements, and how the individual actuators should be coordinated to obtain desired motion at the robot end-effecter. These are questions central to the design and control of robot mechanisms. To begin with, we will restrict ourselves to a class of robot mechanisms that work within a plane, i.e. Planar Kinematics. Planar kinematics is much more tractable mathematically, compared to general three-dimensional kinematics. Nonetheless, most of the robot mechanisms of practical importance can be treated as planar mechanisms, or can be reduced to planar problems. General three-dimensional kinematics, on the other hand, needs special mathematical tools, which will be discussed in later chapters. 4.1 Planar Kinematics of Serial Link Mechanisms Example 4.1 Consider the three degree-of-freedom planar robot arm shown in Figure 4.1.1. The arm consists of one fixed link and three movable links that move within the plane. All the links are connected by revolute joints whose joint axes are all perpendicular to the plane of the links. There is no closed-loop kinematic chain; hence, it is a serial link mechanism. x End Effecter Joint 1 Link 3 Link 2 Link 1 Joint 3 Joint 2A O 2 1 y 1θ Link 0 3 eφ3θ 2θ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ e e y x B E Figure 4.1.1 Three dof planar robot with three revolute joints To describe this robot arm, a few geometric parameters are needed. First, the length of each link is defined to be the distance between adjacent joint axes. Let points O, A, and B be the locations of the three joint axes, respectively, and point E be a point fixed to the end-effecter. Then the link lengths are EBBAAO === 321 ,, . Let us assume that Actuator 1 driving docsity.com link 1 is fixed to the base link (link 0), generating angle 1θ , while Actuator 2 driving link 2 is fixed to the tip of Link 1, creating angle 2θ between the two links, and Actuator 3 driving Link 3 is fixed to the tip of Link 2, creating angle 3θ , as shown in the figure. Since this robot arm performs tasks by moving its end-effecter at point E, we are concerned with the location of the end-effecter. To describe its location, we use a coordinate system, O-xy, fixed to the base link with the origin at the first joint, and describe the end-effecter position with coordinates e and e . We can relate the end-effecter coordinates to the joint angles determined by the three actuators by using the link lengths and joint angles defined above: x y )cos()cos(cos 321321211 θθθθθθ +++++=ex (4.1.1) )sin()sin(sin 321321211 θθθθθθ +++++=ey (4.1.2) This three dof robot arm can locate its end-effecter at a desired orientation as well as at a desired position. The orientation of the end-effecter can be described as the angle the centerline of the end-effecter measured from the positive x coordinate axis. This end-effecter orientation eφ is related to the actuator displacements as 321 θθθφ ++=e (4.1.3) The above three equations describe the position and orientation of the robot end-effecter viewed from the fixed coordinate system in relation to the actuator displacements. In general, a set of algebraic equations relating the position and orientation of a robot end-effecter, or any significant part of the robot, to actuator or active joint displacements, is called Kinematic Equations, or more specifically, Forward Kinematic Equations in the robotics literature. Exercise 4.1 Shown below in Figure 4.1.2 is a planar robot arm with two revolute joints and one prismatic joint. Using the geometric parameters and joint displacements, obtain the kinematic equations relating the end-effecter position and orientation to the joint displacements. End Effecter Joint 1 Link 3 Link 2 Link 1 Joint 3 Joint 2 y 3 1θ eφ 3θ x ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ e e y x d E B O A Link 0 Figure 4.1.2 Three dof robot with two revolute joints and one prismatic joint docsity.com Note that the right hand sides of the above equations are functions of eee yx φ,, alone. From these wrist coordinates, we can determine the angle α shown in the figure.1 w w x y1tan−=α (4.2.2) Next, let us consider the triangle OAB and define angles γβ , , as shown in the figure. This triangle is formed by the wrist B, the elbow A, and the shoulder O. Applying the law of cosines to the elbow angle β yields 2 21 2 2 2 1 cos2 r=−+ β (4.2.3) where , the squared distance between O and B. Solving this for angle 222 ww yxr += β yields 21 222 2 2 11 2 2 cos ww yx −−+−=−= −πβπθ (4.2.4) Similarly, 2 21 2 1 2 cos2 =−+ γrr (4.2.5) Solving this for γ yields 22 1 2 2 2 1 22 11 1 2 costan ww ww w w yx yx x y + −++ −=−= −−γαθ (4.2.6) From the above 21,θθ we can obtain 213 θθφθ −−= e (4.2.7) Eqs. (4), (6), and (7) provide a set of joint angles that locates the end-effecter at the desired position and orientation. It is interesting to note that there is another way of reaching the same end-effecter position and orientation, i.e. another solution to the inverse kinematics problem. Figure 4.2.2 shows two configurations of the arm leading to the same end-effecter location: the elbow down configuration and the elbow up configuration. The former corresponds to the solution obtained above. The latter, having the elbow position at point A’, is symmetric to the former configuration with respect to line OB, as shown in the figure. Therefore, the two solutions are related as γθθθθφθ θθ γθθ 22''' ' 2' 23213 22 11 −+=−−= −= += e (4.2.8) Inverse kinematics problems often possess multiple solutions, like the above example, since they are nonlinear. Specifying end-effecter position and orientation does not uniquely determine the whole configuration of the system. This implies that vector p, the collective position and orientation of the end-effecter, cannot be used as generalized coordinates. The existence of multiple solutions, however, provides the robot with an extra degree of flexibility. Consider a robot working in a crowded environment. If multiple configurations exist for the same end-effecter location, the robot can take a configuration having no interference with 1 Unless noted specifically we assume that the arc tangent function takes an angle in a proper quadrant consistent with the signs of the two operands. docsity.com the environment. Due to physical limitations, however, the solutions to the inverse kinematics problem do not necessarily provide feasible configurations. We must check whether each solution satisfies the constraint of movable range, i.e. stroke limit of each joint. E x End Effecter A’ Wrist β 1θ y 3θ 2θ B O A '1θ '3θ γ Elbow –Down Configuration '2θ Elbow-Up Configuration Figure 4.2.2 Multiple solutions to the inverse kinematics problem of Example 4.2 4.3 Kinematics of Parallel Link Mechanisms Example 4.3 Consider the five-bar-link planar robot arm shown in Figure 4.3.1. 2211 2211 sinsin coscos θθ θθ += += e e y x (4.3.1) Note that Joint 2 is a passive joint. Hence, angle 2θ is a dependent variable. Using 2θ , however, we can obtain the coordinates of point A: 2511 2511 sinsin coscos θθ θθ += += A A y x (4.3.2) Point A must be reached via the branch comprising Links 3 and 4. Therefore, 4433 4433 sinsin coscos θθ θθ += += A A y x (4.3.3) Equating these two sets of equations yields two constraint equations: docsity.com 44332511 44332511 sinsinsinsin coscoscoscos θθθθ θθθθ +=+ +=+ (4.3.4) Note that there are four variables and two constraint equations. Therefore, two of the variables, such as 31,θθ , are independent. It should also be noted that multiple solutions exist for these constraint equations. Joint 2 End Effecter Joint 1 Link 4 Link 3 Point A Joint 5 Link 2 Joint 4Link 1 Joint 3 4θ 1θ 3θ 2θ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ e e y x y AA yx , x Link 0 Figure 4.3.1 Five-bar-link mechanism Although the forward kinematic equations are difficult to write out explicitly, the inverse kinematic equations can be obtained for this parallel link mechanism. The problem is to find 31,θθ that lead the endpoint to a desired position: . We will take the following procedure: ee yx , Step 1 Given , find ee yx , 21,θθ by solving the two-link inverse kinematics problem. Step 2 Given 21,θθ , obtain . This is a forward kinematics problem. AA yx , Step 3 Given , find AA yx , 43,θθ by solving another two-link inverse kinematics problem. Example 4.4 Obtain the joint angles of the dog’s legs, given the body position and orientation. docsity.com