Force/motion sliding mode control of three typical mechanisms

This paper proposes a sliding mode control (SMC) algorithm for trajectory tracking of the slider‐crank mechanism, quick‐return mechanism, and toggle mechanism. First, the dynamic models suitable for the controls of both the motion and constrained force are derived using Hamilton's principle, the Lagrange multiplier, and implicit function theory. Second, the SMC is designed to ensure the input torques can achieve trajectory tracking on the constrained surfaces with specific constraint forces. Finally, the developed method is successfully verified for effectiveness of the force/motion controls for these three typical mechanisms from the results of simulation. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

new sliding surface in terms of motion error and force error, and claimed that the errors in [11] are improved; therefore, the asymptotic stability of the motion-tracking error and forcetracking error can be ensured. However, Dixon and Zergeroglu [15] pointed out an error in the sliding mode control stability analysis of [14]. In this chapter, our intent is to improve the errors in [11,14] and simplify the control design and stability proof for the three typical mechanisms, including the slider-crank mechanism, the quick-return mechanism and the toggle mechanism as shown in Figs. 1~3 respectively, which are not seen in any references addressing the force/motion SMC. Here, a separate sliding surface is proposed using the measurements of the angular position and speed of the crank, but the SMC algorithm is derived as well in a simple manner using only the force tracking error to construct the controller. In these schemes, the force tracking error is shown to be arbitrarily small by changing the force control feedback gain. Then, by exploiting the structure of its dynamics, the fundamental properties of the dynamics are obtained to facilitate controller design, whereby the asymptotic stability of motion tracking error in sliding surface and force tracking error accumulated in controller can be ensured. The organization of this chapter is arranged as follows. In Section 2, the kinematic and dynamic analysis of the multi-body mechanism is investigated. A number of previous papers [4][5][6][7][16][17] have shown the position and speed controllers for the regulation and tracking problems of the multi-body mechanism in the theoretical analysis and experimental results. However, control of the constrained force has not been investigated. The SMC laws are designed in Section 3. The simulated examples are shown in Section 4 and, finally, some conclusions are drawn.

Dynamics analysis 2.1 Dynamic equation of motion
Based on the Euler-Lagrange formulation [4], the equation of motion for a mechanism can be expressed as: is the partial derivative of the constraint equation with respect to the coordinate and is called the constraint Jacobian matrix, is the vector of nonconservative forces and n R ∈ U is the vector of applied control efforts. In order to obtain the general form of the force/motion controller design, we rewrite the nonlinear vector as: www.intechopen.com

Dynamic properties of the mechanism
Equation (3) is similar to the motion equation of an n-link rigid constrained robot [11,15] in the state space. Two simplifying properties should be noted about this dynamic structure: Property 1. The individual terms on the left-hand side of Equation (3) and the whole dynamics are linear in terms of a suitably selected set of equivalent manipulator and load parameters, i.e., where Y(Q,Q,Q) is a nr × matrix; r R α∈ is the vector of equivalent parameters. Due to the presence of m constraints, the degree of freedom of the mechanism is (n-m). In this case, (n-m) linearly independent coordinates are sufficient to characterize the constrained motion. From the implicit function theorem, the constraint Equation (1) can always be expressed as [18]: Equation (5) is assumed that the elements of q are chosen to be the last (n-m) components of Q. If not the above case, Equation (1) still could always be reordered so that the last (n-m) equations would correspond to q and the first m equations to p. That is, .
Then, to simplify the equation form of the dynamic model, defining and using Equation (5), we have: Therefore, the dynamic model of Equation (3) restricted to the constraint surface can be expressed in a reduced form as: where =+ 1C N (q,q) M(q)L(q) N (q,q)L(q) .
By exploiting the structure of Equation (9), three properties can be obtained as follows: Property 3. In terms of a suitably selected set of parameters, the motion equation (9) is still linear, i.e. 1 ) + += α 1G M(q)L(q)q N (q,q)q N (q Y (q,q,q) .
www.intechopen.com Property 4. Define the matrix as L (q)Φ 0 . The above three properties are basic principle in designing the force/motion SMC law.

The sliding mode controller design
A number of previous papers have only shown the position and speed controller designed for the regulation and tracking problems control of the constrained mechanisms. However, control of the constrained force has not been investigated in the previous studies. In this section, a separate sliding surface is proposed using the measurements of the angular position and speed of the crank, but the SMC algorithm is derived as well in a simple manner using only the force tracking error to construct the controller. Given a desired trajectory d q and a desired constrained force d F , or identically a desired multiplier d λ , which satisfy the imposed constraint, i.e., control objective is to determine the SMC law such that → d qq and → d λ λ as t →∞.
From the SMC methodology, we define the tracking error where nm R − ∈ r q is the reference trajectory and is a tunable matrix. The sliding controller [12] is defined as: where 1 Y is a nr × matrix of known functions of q,q and q , L(q) is defined in Equation (6), is the vector of switching functions, and is a force control that is defined as: where K is a m m × constant matrix of force control feedback gains, and λ e is the error vector of the multipliers and defined as
Defining α as a constant r-dimensional vector and replacing q by the reference trajectory r q , then the linear parameterization of the dynamics (Property 3) leads to: Gr 1rrr M ( q) L ( q) q N( q, q) q N ( q) Y( q, q, q) .
Using the derivative of the sliding surface equation (14) and substituting into Equation (11), we obtain: Then combining Equation (18) with Equation (20) and using Equation (19), we obtain: According to property 5, the above equation becomes: To derive the control algorithm, the generalized Lyapunov function is considered as: Differentiating V with respect to time and using property 4, Equation (23) becomes: The ϕ is chosen as: such that 0 ≤ <

TT 11
Vs L L s .
In the derivation of Eq. (24), it is noted that ( ) 2 ( , ) is also skew-symmetric, which is the same as those in [11,14]. Besides, the special cases of the three typical mechanisms in this chapter, ( 2 ) is always equal to zero for 1 nm −=.
To reduce the chattering phenomenon along the sliding surface 0 s = , we adopt the quasilinear mode controller [13], which replaces the discontinuous term of sign function of www.intechopen.com Equation (25) with a continuous function inside a boundary layer around the sliding surface [24]. Therefore, the sgn(S) is replaced by the saturated function: where ε is the width of the boundary layer. This limits the tracking error and guarantees an accuracy of ε order while alleviating the chattering phenomenon.
From Equation (23) and Equation (26), it is evident that a sliding surface 1 s is at last converged exponentially to zero, i.e., → m e0 as t →∞. As if → d qq , the condition = dd p σ(q ) also implies that → d pp . Therefore,

Simulation examples of the three typical mechanisms 4.1 The slider-crank mechanism
For more details on the kinematic and dynamic analysis of the slider-crank mechanism, refer to [19]. Using Hamilton's principle and Lagrange multipliers [20] and adopting the generalized coordinate vector in Equation (1) for the slider-crank mechanism shown in Figure 1, the dynamic equation can be obtained associated with the following matrices and elements: where the dimensions of the slider-crank mechanism are 2, 1 nm = = , and 1 r = in the dynamic analysis. For the single degree-of-freedom slider-crank mechanism, only one constraint equation exists, which can be shown as: www.intechopen.com The position of the slider B can be expressed as: Substituting Equation (28) into Equation (29) yields: The angular displacement of the crank can be obtained as: The result can also be obtained if the cosine law is applied. The Jacobian matrix of the constraint equation (28) is Differentiating Equation (28) with respect to time yields the constraint velocity equation: Therefore, the matrix defined in Equation (6) becomes and its first time derivative becomes 22 sin cos cos sin 0 cos The dynamic equation (9) of the slider-crank mechanism, when restricted to the constraint equation (28), can be expressed as: 12 Using Equation (19), the applied control effort can be derived as: where For numerical simulations, the parameters of the slider-crank mechanism are chosen as: All the parameters in the SMC controller are chosen to achieve the best transient performance in numerical simulations under the limitation of the control effort and the requirements of stability. Furthermore, for the reason of using a single input actuation on joint 1, the control effort is only needed in the second equation of the constrained motion of Equation (36). As to the first part of Equation (36), it shows that the force equilibrium either with holonomic or nonholonomic constraints and the torque is exerted at joint 2. The responses of the crank angle, which is shown in Figure 5(a), reach the desired value in about 0.8 sec. The slider position manipulated from Equation (29) is shown in Figure 5(b). The tracking results of the crank angle θ and the slider position B x coincide with previous studies by Fung et al. [16,17]. The control effort of the applied torque τ is shown in Figure   5(c) and the sliding surface 1 s is shown in Figure 5(d). The Lagrange multiplier C λ is shown in Figure 6(a), and constraint forces of joints 1 and 2 are shown in Figure 6(b) and Figure 6(c), respectively. From Figures 5 and 6, the control objectives of force/motion of the slider-crank mechanism are achieved successfully.

The quick-return mechanism
To present the robustness and a well-established control method of the SMC controller, the quick-return and toggle mechanisms (see Section 4.3) will be chosen to verify the SMC algorithm, which is then adequately developed to the general case of multi-body mechanisms. The quick-return mechanism is addressed first, where the kinematic and www.intechopen.com dynamic analysis of the mechanism is found in [21] and the generalized coordinate vector in Equation (1) for the quick-return mechanism shown in Figure 2 is adopted. The dynamic equation can be obtained and is associated with the following matrices and elements: cos cos sin sin 0 sin sin cos cos sin cos 0 where ϕ can be obtained by analyzing its geometric relations 1 sin tan cos The position of slider C can be expressed as: The Jacobian matrix of the constraint equations is: Therefore, the matrix defined in Equation (6) Figure 7(d). The Lagrange multiplier C λ is showed in Figure 8(a). The constraint forces of joints 1-3 are shown in Figures 8(b)-(d), respectively. From Figures 7 and 8, the control objectives of force/motion of the quick-return mechanism are achieved successfully.

The toggle mechanism
For more details of the kinematic and dynamic analysis of the toggle mechanism, refer to where φ can be obtained by analyzing its geometric relation as: Therefore, the matrix defined in Equation (6) The parameters 11 21 ,,, , , , ,,, ,,, , in the simulations. Furthermore, in order to show that the SMC is insensitive to parametric variation, the effects of friction forces in joints are considered in this toggle mechanism system by using the Lagrange multiplier method. The comparisons between the nominal case without considering friction forces and the case with friction forces are shown in Figures 9-11. Figures 9(a)-(c) show the trajectories of angles 1 θ , 2 θ and 5 θ , respectively. Figures 10(a)-(b) show the positions of sliders B and C, respectively. Figures 10(c)-(d) illustrate the control effort τ and the sliding surface 1 s , respectively. Finally, Figure 11(a) shows the Lagrange multiplier C λ and Figures 11(b)-(d) address the constraint forces 1 f , 2 f and 3 f acting on the joints 1, 2, and 3, respectively.
From the numerical results, it is found that the control efforts τ are almost identical for both cases whether the friction forces are considered or not. From the above figures, the www.intechopen.com force/motion control objective of a toggle mechanism using the SMC is achieved successfully and the system responses are insensitive to the effects of friction forces.

Conclusion
Based on the Lyapunov theorem, we successfully derived a generalized SMC algorithm in a simple manner. The algorithm used tracking the Lagrange multiplier error to facilitate controller design and proposed a separate sliding surface in terms of the displacement and velocity. Furthermore, some properties of the dynamic structure were presented and used to reduce the dynamic model equations. Finally, the slider-crank, quick-return, and toggle mechanisms were employed to illustrate and verify the methodology developed. From the numerical results, we conclude that the effectiveness in application of the developed SMC method is successfully verified in regards to the force/motion controls for these three typical mechanisms. First, fast attainment of the control objective: in the simulations, the three typical mechanisms reached the desired value in less than 1 second. Second, no overshoot in the control process: from the numerical results, the force/motion control objectives of the three mechanisms using the SMC were achieved successfully and the system responses did not overshoot in the whole control process. Third, insensitivity to parametric variation: the control efforts τ are almost identical and the system responses are insensitive to the effects of the friction forces.

Acknowledgment
The      The main objective of this monograph is to present a broad range of well worked out, recent application studies as well as theoretical contributions in the field of sliding mode control system analysis and design. The contributions presented here include new theoretical developments as well as successful applications of variable structure controllers primarily in the field of power electronics, electric drives and motion steering systems. They enrich the current state of the art, and motivate and encourage new ideas and solutions in the sliding mode control area.

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