Target Tracking Using Double Pendulum
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1 Target Tracking Using Double Pendulum Brian Spackman 1, Anusna Chakraborty 1 Department of Electrical and Computer Engineering Utah State University Abstract: This paper deals with the design, implementation and simulation of a linear controller for a double link pendulum which controls its chaotic motion in free fall and guides it to hit a stationary target as well as track a time-varying trajectory. The paper also compares the performance of the linear controller utilizing either a single motor or two motors. The simulation setup considers a double link pendulum in the x-y plane along-with all the targets. The controller design is based on the concept of Linear Quadratic Regulators (LQR) and is tuned to hit 4 different targets during a single run. I. INTRODUCTION The chaotic motion of a double pendulum has been an active area of research for the past few decades. Different models have been designed so that the chaotic motion can be converted into sustained oscillations which can be expressed in terms of simple quadratic equations mathematically. However, the practical applications of the double pendulum have increased extensively in the last decade. Identification that human arm motion is equivalent to that of a controlled double pendulum has been a major breakthrough in the manufacture of prosthetic technology. However since the double pendulum is a highly nonlinear system, the cost of controlling these limbs with sophisticated controllers has increased the cost of these prosthetics dramatically[1]. Specially, the prosthetics of athletics require highly complex control and are very expensive to design and manufacture. Although research is ongoing regarding the design of low cost controllers, no ground-breaking solution has yet been achieved[2]. Another major application of the double pendulum is that of load-lifting cranes. These are used in most fields and control of the load at the end of the second arm is very essential for safety of the payload and ground personnel. Safety is the main reason that these cranes are not automated. This is because to achieve smooth control of the crane with load requires very intricate and complex designing and hence it is not at all cost effective. In this paper, we have shown that the non-linear system can be expressed in terms of linear equations and can also be effectively controlled with the help of linear controllers. The paper also talks about designing the controllers to achieve a critically damped system and attempts to show the balance between the smooth control and the input voltage. The paper has been modelled in the following manner. Section II talks about the equations of motion and also the mathematical calculations required for modelling the system. Section III discusses the simulation set-up and LQR controller. Section IV talks about the results obtained from the experiment and compares the graphical results. Section V summarizes the experimental results and conclusions drawn and section VI discusses the possible future scope of this paper. II. MODELLING A DOUBLE PENDULUM A. SYSTEM DYNAMICS Fig 1. Model of the Double Pendulum The following equations are derived for a system with two motors one motor is at the base of the first arm of the pendulum and the second motor is at the link between the first and the second arm. While designing the double pendulum model we have assumed that
2 both arms of the pendulum are of equal length so that the entire model has a greater reachability [5]. Here: (1) (2) (3) (4) Where (14) (15) (16) (17) (18) The Lagrangian function is defined as: (5) (6) (7) (19) The state space representation of the double link pendulum with a single motor is as follows: The equations of motion are obtained from the following set of equations: (8) (9) B. DETERMINATION OF ANGLE OF IMPACT (20) Where E 1 and E 2 are the motor voltages and θ 1 and θ 2 are the degrees of freedom for the system. For the design with the single motor attached at the base of the first arm of the pendulum equation (9) gets modified as follows: (10) (0,0) (x1,y1) After simplification of equations (8) and (9) we get: (11) (12) The state space representation of the double link pendulum with two motors is as follows [6]: (13) Fig 2. Location of Target with respect to the pendulum Defining the ground location as (0,0) and the target location as (x1,y1), a circle with radius l can be drawn around each location and the intersecting points are the location where the first bob must be located in order for the second bob to reach the target location. For most scenarios within reach, there are two such intersections defined as x and y below. The equations for the two circles are written below. x 2 + y 2 = l 2 (21)
3 (x x 1 ) 2 + (y y 1 ) 2 = l 2 (22) Setting these equal to each other and solving for x yields the following equation. x = x 1 2 2y 1 y+y 1 2 2x 1 (23) This can be plugged back into the first equation and simplified into a second degree polynomial. (4y x 1 2 )y 2 + ( 4y 1 3 4x 1 2 y 1 )y + (x y x 1 2 y 1 2 4x 1 2 l 2 ) = 0 (24) Where (29) = output energy (30) =input energy (31) This is solved by the quadratic formula. Plugging that solution back into the original equation allows one to solve for x. Both are solved below, representing four locations where the first bob may be located. y = 4y 1(x y 1 2 ) ± 4x 1 4l 2 (x y 1 2 ) (x y 1 2 ) 2 8(x y 1 2 ) x = ± l 2 y 2 (25) Only two of these can be intersect locations. To determine which two of the four sets are intersect locations, they are each used in the distance equation below to determine which points are located at a distance of l away from the target. D 2 = (x x 1 ) 2 + (y y 1 ) 2 (26) Angles θ 1 and θ 2, the angles made by the first and second arm respectively, are determined by trigonometric functions. θ 1 = tan 1 ( y x ) (27) θ 2 = tan 1 ( y 1 y x 1 x ) (28) III. SIMULATION DESIGN A. CONTROLLER DESIGN In this paper the Linear Quadratic Regulator Controller (LQR) has been used to control the arms of the pendulum. The theory of optimal control is concerned with operating a dynamic system at minimum cost [7]. For a continuous linear tie invariant system x = Ax + Bu and y = Cx where u = Fx, the objective is to minimize the cost function which is defined as follows: The LQR controller is desired to reduce both these energies since they both contribute to the cost function. However, decreasing the energy of output requires a large control input and a small control input leads to large outputs. So the positive definite weighting matrices Q and R establishes a trade-off between these conflicting parameters. When we used a single motor in the system, to control the pendulum, we initially checked the controllability of the system to determine whether it is even possible to control the system in the desired manner with a single motor. The controllability of the system is defined as: (32) The B matrix used for this check is that shown in equation (20). The rank of the matrix was 4, meaning the system was controllable with only one motor. For the single motor case, we defined our Q matrix such that we did not have any control over the velocity of the second arm. The second arm was completely dependent on the first arm s velocity to set to the desired angle and hit the target. The beginning guess for weights on each parameter were based on Bryson s rule and then further tuned to match the requirements of the system. For the double motor scenario, our Q matrix had weightage on all the four parameters that is the angles of the first arm and second arm and the velocities of the two arms. B. SIMULATION SET-UP The block diagram of the entire system is as below:
4 Fig 3. Block Diagram of the system The simplified diagram above represents the system operation. Initially we consider that there is some noise in the system. The Plant represents the system. The outputs from the plant, that is the 4 state variables, are passed onto the LQR controller which is in a negative feedback loop with the plant. The controller generates the values for K which are the system gains and we check the outputs, that is the angles of the two arms, and we also record the control input to show the difficulties in balancing the two responses. The Q and R matrices used in the system and the various graphs obtained are all discussed in Section IV of this paper Fig 4. Fluctuation of Θ 1 with change of Targets IV. EXPERIMENTAL RESULTS A. OUTPUT WITH SINGLE MOTOR Our aim was to create a system that was critically damped without any transience in the output. Since our focus was on the output terms rather than the control input we will see in the following graphs that the input is extremely high and often has large transience which is quite undesirable. However, with the single motor scenario, despite repeated tunings we were unable to decrease the oscillation in the system beyond a certain limit. Although the system settles down very quickly, the peaks of the oscillations are quite high and the system was not critically damped with a single motor. CASE 1: The weighting matrices used in Case 1 are: Fig 5. Fluctuation of Θ 2 with change of Targets Fig 6. Fluctuation of Input Voltage E1with change of Targets
5 The huge oscillations of Θ 1 and Θ 2 can be observed in the above graphs although these Q and R matrices are completely tuned. Also the huge oscillations of the input voltage can be seen in figure (5). The oscillations reach up to 300 volts in the beginning which is completely unacceptable. CASE 2: The weighting matrices used in this case are: Fig 9. Fluctuation of Input Voltage E 1with change of Targets Fig 7. The responses obtained are as follows: Fluctuation of Θ 1 with change of Targets Now we put more weight on the control input and hence we can observe from the figure that although the transience is present the peaks are much lower than the previous case. The maximum peak is at 175 volts as opposed to the 300 volts in the previous case. However the main trade off is in the transience of the angles Θ 1 and Θ 2. The oscillations have increased significantly and continue far longer than the previous case. This clearly reflects that the balance between the input control voltage and the output angles is quite subtle. But from this scenario of single motor we concluded that it is not possible to reduce the transients any further than case 1. Although the goal of controlling a non-linear system with a linear controller has been achieved, the system is not damped and hence when practically implemented will cause significant problems to the motor. B. OUTPUTS WITH DOUBLE MOTORS Now we use two motors to control both the arms of the pendulum separately. Under this scenario also we describe two cases with different weights on the control input that will demonstrate quite vividly balance between the deviation from a desired angle and the input voltage required to get there. CASE 1: The weighting matrices used in Case 1 are: Fig 8. Fluctuation of Θ 2 with change of Targets
6 The controllability of the system was checked again and the controllability matrix had a full rank of 4. The response graphs are below: Fig 12. Targets Fluctuation of Input Voltage E1 and E2 with change of Figures (9) and (10) show the fluctuations of Θ 1 and Θ 2. In this case we find that except for very small oscillations in Θ 1 the system is critically damped with no unwanted transience. This gives complete smooth control on the two arms. However, the control input oscillations for both E 1 and E 2 are quite high. This is because we have put negligible weightage on the control input. CASE II: In this case we will put quite a high weight on the control input. The result will be decreased peaks of oscillations in the control input voltages. But oscillations in the output angles increase considerably. Fig 10. Fluctuation of Θ 1 with change of Targets Fig 13. Fluctuation of Θ 1 with change of Targets Fig 11. Fluctuation of Θ 2 with change of Targets
7 Fig 14. Fluctuation of Θ 2 with change of Targets In this trajectory the red dots represent the trajectory and the black dots represents the path followed by the pendulum in tracking the trajectory. We see that the pendulum follows the path quite efficiently except at the point where the trajectory makes a sharp turn. V. CONCLUSION In this paper we have used a linear (LQR) controller and successfully controlled the chaotic motion of a double pendulum which is highly non-linear. From the experimental results it is quite clear that the use of two motors is much more effective in controlling the motion and reaching the target than a single motor. The trade-off between the input energy and the output energy is always present for an LQR controller and the weights of the Q and R matrices will vary depending on the requirement. In our case, we desired a critically damped system with minimum transience in the output and the LQR controller performs very well in that respect. For the moving trajectory we find that the pendulum tracks the moving trajectory with a small amount of time lapse. Except at sharp edges and corners where the pendulum has to adjust it s angle, overall the pendulum tracks the time-varying trajectory quite well. VI. FUTURE SCOPE Fig 15. Targets Fluctuation of Input Voltage E1 and E2 with change of C. TIME VARYING TRAJECTORY In this case we will look at how well the double pendulum tracks a time varying trajectory. In order to have an autonomous crane which is cost effective and very efficient in lifting loads and following fixed trajectories, we introduced the idea of using a linear controller to control the system dynamics. So, as a continuation of this work, a considerable load may be attached at the end of the second arm of the pendulum and the equations of motion have to be modified to take that load into account. The control of that model will give an exact idea of the practical implementation of an autonomous crane with load. Another aspect that can be worked upon is the implementation of a smart controller which can calculate by itself the angle by which it has to turn the least during tracking. In that way the time lapse can be reduced by a big amount. This can be achieved by calculating the difference between the angle the pendulum has to move and the angle that the pendulum is currently at in both directions and whichever angle is smaller will be the more efficient choice for the pendulum. Fig 16. The Pendulum tracking a Time Varying trajectory
8 VII. REFERENCES [1] Wentink, EC; Koopman, HFJM; Stamigioli, S; et al. Variable Stiffness Actuated Prosthetic Knee to Restore Knee Buckling Stance: A Modeling Study. Medical Engineering & Physics, Volume 35, Issue 6, Pages Jun [2] Yoyo Au. Double Inverted Pendulum Control. [3] Troy et al, Chaos in Double Pendulum, Chaos_in_a_double_pendulum-AJP.pdf [4] Google.com [5] Google.com [6] Google.com [7] Linear Systems Theory, Joao. P. Hespanha, Lecture 10: Preview of Optimal Control
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