Solving the Kinematics of Planar Mechanisms. Jassim Alhor

Transcription

1 Solving the Kinematics of Planar Mechanisms Jassim Alhor

2 Table of Contents 1.0 Introduction Methodology Modeling in the Complex Plane Writing the Loop Closure Equations Solving the Loop Closure Equations Isotropic Coordinates Reduction to Bilinear Quadratics Solution of Bilinear Equations Conclusions 7 Referances 8

3 1.0 Introduction This case study illustrates a general method to analyze planar mechanisms, consisting of rigid links and connected by rotational and/or translational joints. This method can be used either to analyze the motion of a given mechanism or as a step in the design of a mechanism to produce a desired motion. Both input/output problems and path generation equations are treated. The goal of this method is to determine the motion of the links as one or more input links are displaced. Common applications are: Function generator mechanisms: the interest is in the motion of a single output joint as a function of a single input motion. Path generating mechanisms: the path of the mechanism in the plane is of some importance. Robotics: the motion of an end-effector link in response to multiple input motions is of interest. 2.0 Methodology This method could be divided into three steps. After describing the links as vectors in the complex plane, a simple recipe is outlined for formulating a set of polynomial equations, which determine the locations of the links when the mechanism is assembled. It is then shown how to reduce this system of equations to a generalized eigenvalue problem, or in some cases, a single resultant polynomial.

4 2.1 Modeling in the Complex Plane The first step is to describe all of the links in vector form in the complex plane. For example, we may equivalently say that the location of A is given by the complex vector a = A x + ia y (see Figure 1). Figure 1: Vector in the Complex Plane Suppose that the link is moved by a t translation and rotation angle θ. The new position of vector A is A = t + θa, where t is the distance between the reference point and vector. Now, the final position of the last point at the end of a series can be expressed using this notation. This would be a useful tool in writing the loop closure equations for the mechanism. 2.2 Writing the Loop Closure Equations The next step is to write the loop closure equations for the mechanism. The loop closure equations could be found by using the paths of two or more links. The

5 kinematic equations for a general planar mechanism, having l loops and only rotational joints, consist of l equations, each of the form Where: n 1 + j= 1 a o a j Θ j = 0 Equation (1) a j : complex vectors describing the links in their reference positions. n : number of links Θ j : e iθj, where θ j is in radians l : number of kinematic loops 2.3 Solving the Loop Closure Equations The Third step is to find all solutions for the loop closure equations. This is done by selecting some joint as the primary output joint and eliminating all other joint variables from the closure equations, leaving only a single polynomial equation. The following steps should be followed to the single polynomial equation Isotropic Coordinates The usual approach to equations of the form considered is to take their real and imaginary parts, in which case Θ j becomes cosθ j + isinθ j. Equations in the conjugate variables Θ j * are generated by simply taking the complex conjugates of the closure equations. This gives l equations of the form n 1 + j= 1 a o * a j * Θ j * = 0 Equation (2) Where * indicates conjugation.

6 Θ j and Θ j * are treated as independent variables with constraint that for all rotational joints Θ j Θ j * = 1, j = 1,, 2l Equation (3) Reduction to Bilinear Quadratics This step will eliminate half of the variables. The closure equations (1) can be solved to express l of the variables Θ j as linear combinations of the others. The same is done for the conjugate variables Θ j * using equations (2). Substitution into the unit vector equations (3) leaves the following system of 2l equations ( b oj l + bkjθ k )( boj * + b k = 1 l k = 1 kj * Θ k *) = 1 Equation (4) Solution of Bilinear Equations By replacing Θ j * with Θ j -1 for j=1,, l-1, we get equations of the form Θ l Θ l * - 1 = 0 Equation (5) ( b oj + l k = 1 b kj Θ k )( b oj * + l 1 k = 1 b kj * Θ 1 k + b lj * Θ i *) = 1 Equation (6) After multiplying each of the equations (5) and (6) by each of the monomials of degree l-1, we get the expanded set of equations in matrix form Qm = (Q 1 +Q 2 Θ l *)m = 0 Equation (7) where m is a column vector of length (l+1)( 2l-1 l ), and Q 1 and Q 2 are square.

7 The value of Θ l * can be found using the following equation det (Q 1 +Q 2 Θ l *) = 0 Equation (8) which is a polynomial in Θ l * of degree at most ( l 2l-1 ). 3.0 Conclusion A general method was illustrated to analyze planar mechanisms, consisting of rigid links and connected by rotational and/or translational joints. The solutions must be tested for physical validity. This is analogous to the more familiar situation of equations with real coefficients in which the physically meaningful solution must be pure real (zero imaginary part). In this case, it is required that Θ j Θ j *=1, that is, the rotation vectors must have unit magnitude.

8 References Erdman, Arthur G., and Sandor, George N., Mechanism Design. Prentice Hall, New Jersey, Wampler, C. W., Solving the Kinematics of Planar Mechanisms. ASME Journal of Mechanical Design, Vol. 121, pp , September Wilson, Charles E., and Sadler, J. Peter, Kinematics and Dynamics of Machinery. HarperCollins College Publishers, New York, 1993.