Abstract
Dynamic equations and the control law for a class of robots with elastic underactuated MIMO system of legs, athlete Robot, are discussed in this paper. The dynamic equations are determined by Euler-Lagrange method. A new method based on hierarchical sliding mode for controlling postures is also introduced. Genetic algorithm is applied to design the oscillator for robot motion. Then, a hierarchical sliding mode controller is implemented to control basic posture of athlete robot stepping. Successful simulation results show the motion of athlete robot.
1. Introduction
Two-legged robot is an interesting topic which lasts for long time [1–3]. One of the classical forms of this type, humanoid robot, which only has solid links, can move or balance through ZMP method [4–6]. Anyway, this method is only appropriate with MIMO system which has the same number of inputs and outputs. One of the other disadvantages of ZMP is that the robot moves very slowly and unflexibly. In order to improve the flexibility of robot and energy saving capabilities, some authors [7–13] put elastic components for smooth dynamic motion. By these researches [7–13], robot is flexible when the feet of robot still exist. One approach suggested [14, 15] is to replace the leg and foot by an elastic leg. This approach makes the robot become the underactuated MIMO system, called athlete robot (AR). The AR robot study is based on the idea of elastic legs for disabled people (Figures 1 and 2). In the case of AR, solid legs are replaced by elastic legs and two torques are substituted by one torque (Figure 3). Ryuma Niiyama and his colleagues studied this class of robots but their efforts were focused on the experimental results and real biomechanical structure. The dynamic equations and control algorithm were not analyzed. This paper will present the dynamic equation of AR.
Classical methods of ZMP control for humanoid robot [4–6] become useless for ARs due to structure of AR as a MIMO underactuated system. Some researches [16–18] simplified the complicated structure of two-legged robot into simple form: spring-load inverted pendulum (SLIP). Anyway, the SLIP model is not completely equivalent to the former MIMO model. Hence, the controller which is based on the former complicated model will be more reliable. Beside the opinion of control methods for MIMO underactuated nonlinear system [19], this paper also presents new idea that three proportional-controllers (P-controllers) are designed to transform AR to a SIMO system. Hence, it will be more convenient to use multiple control methods for SIMO underactuated system [20–23]. A basic method of hierarchical sliding mode (HSM) control algorithm for SIMO system, which was presented by Qian et al. [20], can balance a SIMO system. Other related works [21–23] used HSM to balance a specific SIMO system. Qian used mathematical methods to prove the stability of sliding surfaces of each layer in [20]. Despite the remarkable contribution in [20], there is still a boundary that control parameters “should” be in. Therefore, based on that boundary, genetic algorithm (GA) is used in our paper to find the appropriate control parameters. Also, GA is also used to design the prescribed trajectory of motion that defines a step of robot.
The paper concludes five sections. The dynamic equations of AR are generated in Section 2. Section 3 infers the mathematical transformation from MIMO underactuated nonlinear structure of AR into a SIMO one. Section 3 also presents application of HSM controller for that robot. Section 4 introduces simulation results. A conclusion in Section 5 ends the paper.
2. Mathematical Model
The elastic legs are designed to be able to self-balance when there is not external force on them (Figure 4) and to accumulate elastic energy for motion. We introduce the following notations:(i): mass center of link (ii): (with ) connecting point of link and link (iii): center of the curved part of elastic leg (iv), : two-edge points of elastic legs with coordinates: and (v): middle point of elastic legs with coordinate: (vi): angle of curved part of elastic legs
Main parameters of AR are described in Figure 5 as follows:(i): mass of ; ; (kg)(ii): length of link ; ; (m)(iii): angle between link 1 and vertical axis, obtaining (rad)(iv): angle between next link and link (rad)(v): inertial moment of link (kgm2)(vi): radius of hunch part (m)(vii): rotational spring coefficient (Nm/rad)
Behavior of elastic leg is determined by Castigliano’s Theorem [24] which provides a good tool for analyzing forces on curved components. AR can be regarded as an equivalent inverted pendulum in Figure 6 where the elastic legs are equivalent to springs.
Consider Figure 6(b) that describes the equivalent model of AR’s leg. The strain potential energy of system can be defined asTotal potential energy of system isKinetic energy of system isLagrange operator isBy using Euler-Lagrange method, dynamic equations will beFrom (5), the dynamic model can be written aswhere , , and the matrices , are calculated by MATLAB/Simulink simulation. New variables are defined to simplify forms of equation (Figure 7).
Relation of variables and is inferred inAlso, denote by and the reference signal of variables and . Consider the angle between one leg and the vertical axis when both legs touch the ground. Sample-time is defined as period time of a step of robot
Many authors [25–27] proposed several solutions for designing the oscillators for the motion of two-legged walking robots. Their methods are very complex and based on intuition and developed by simulations and experiments. In this paper, an algorithm is proposed based on GA.
We consider the reference trajectories of , , , , and as , , , , . Reference trajectories of and are described in Figure 9.
Coordinates and have to be selected through GA.
3. Control Algorithm
In order to use HSC algorithms, new variables will be defined aswhere , .
The dynamic model (6) is rewritten asA simply proportional controller for links 3, 4, and 5 is proposedAlso, definewhere is the second column of matrix .
Substitute (10) and (11) into (9), and obtain By using the conventional change of control inputs (10), the dynamic model (6) is transformed in SIMO models in (12). The AR is a five-order system and it is impossible to calculate directly with just unknown variables due to its complexity and limitation of simulating software (MATLAB/Simulink) (only 25000 characters can appear on MATLAB window). Hence, exact dynamic equations cannot be described visibly. Anyway, simulation process has to be implemented following Figure 10.
From (8) to (12), the error model will beEquation (13) has the form of equation of a SIMO system. Therefore, a solution of using controller which is suitable for high-order SIMO system can be considered. This controller stabilizes variables . This leads to .
Hierarchical sliding mode of structure of hierarchical sliding surfaces is shown in Figure 10.
From the description in Figure 11, sliding surfaces are chosenwhere and .Derivative (16) with respect to time yieldsFrom (14) and (15), we deduce that the th layer SMC comprises the information of layer, and subsystem layer. Hence, define the th layer sliding mode control law asHere, . And , is switching and equivalent control law for th layer. LetDefine Lyapunov function for th layer asDerive (20), and from (17), we obtainDerive (14), and substitute (14) and (18) into new results which is obtained by deriving (14), which yieldsBy considering stability of th layer sliding surface, letFrom (22) and (23), switching control law of th layer can be obtained asand the final control law for controller is selected asFrom general results in [20], we obtain Theorems 1 and 2 below.
Theorem 1. Consider that equations of error of system (9) are described in (13). If control law is chosen as (25) and sliding surfaces are identified as in (16), then is asymptotically stabilized.
Proof. The Lyapunov function of th layer is chosen as in (20). From (23), we obtainIntegrating two sides of formula (21), we obtainConsiderTherefore, we obtainAccording to Barbalat lemma, there existsFrom (30), it means that . Hence, the th sliding surface of is asymptotically stable.
Theorem 2. Consider that equations of error of system (9) are described in (13). If control law is chosen as (25) and th subsystem sliding surfaces are identified as in (14), then is asymptotically stabilized.
Proof. Assume is not asymptotically stabilized when conditions in Theorem 2 are satisfiedHence, from (16), we obtainEquation (32) contradicts Theorem 1. Therefore, we obtainand is asymptotically stabilized.
Theorems 1 and 2 prove the stability of sliding surface. However, the stability of is not guaranteed. Therefore, searching algorithm, such as genetic algorithm (GA), is a solution to find appropriate control parameters in (25). In this case, GA is defined by population of 40 individuals. Each individual contains 24 chromosomes which include values of the following:(i)Control parameters: , , , , , , , , , , , , , , (ii)Trajectory parameters: , , , , , , , (iii)Half-period time of a motion cycle (time for a step of AR): (if AR moves fast, then is chosen in the range: )
The individuals chosen for crossover process have to satisfy the following conditions:(i)All , , and are positive for .(ii).
Fitness function of GA is defined as where .
In GA process, the fitness function of GA is designed to evaluate the results after each loop of searching. If each link tracks trajectory, then and . However, sample time of step is also chosen through GA. Only cannot represent the ability of tracking trajectory in a period of time. Therefore, fitness function should be selected as . The detected result which generates smaller from (34) will be considered as better.
The GA process is shown in Figure 12.
In Figure 12, firstly, 40 random couples were created. In each individual in couples, the structure of gene concludes encoded parameters of control system and reference trajectories (control parameters: , , , , , , , , , , , , , , , , , , , , , , ). We denote a constant value to check if the result is best or not. will be updated if there is better value and the result that makes will be stored as the best result. After crossover couples, 80 new individuals are created. By checking the function of each individual after each simulation, better individuals have more priority to become couples and have more offspring in next generation. Worse individuals have less priority to give their gene to next generation. The loop will continue until we want to stop the program.
4. Simulation Results
System parameters are selected as follows: ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; .
Parameters of HSM controller in (25) are found by GA method as follows: ; ; ; ; ; ; ; ; ; ; ; ; ; ; .
Parameters of reference trajectories in Figure 9 are determined as follows: ; ; ; ; ; ; ; .
Initial values of variables at are specified in Figure 8. Motions of AR under HSM controller are described in Figures 13 and 19.
(a)
(b)
(c)
From initial position (Figure 13(a)), after a period time , AR moves to a new position (Figure 13(b)). An additional time is needed to achieve the exact position of a final sequence. This additional time can be improved by GA techniques. Period time of a step which is will be . Trajectories of each link variables are shown in Figures 14–18.
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(b)
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(e)
(f)
The walking motion of AR for a single step is shown in Figure 19.
From Figures 14 to 18, after 1.12 s, AR finishes a step. However, in the operating process, does not track well the reference trajectories but the final positions are still close to the reference trajectories. Figures 13 and 19 also consolidate that reasoning. The motion of each link affects others significantly. Only motion of link 5 is less affected (Figure 24). Because links 3, 4, and 5 are controlled directly by P-controllers for each link (from (10)), they follow trajectories (Figures 22–24) more closely than link 1 or link 2 does. Anyway, the vibration exists in these figures due to the effect from other links. The response of link 1 and link 2 is not good at first but HSM signal finally leads links 1 and 2 to reference trajectories at the end of period of the step (Figures 20 and 21).
5. Conclusion
In the paper, the authors represent method of generating dynamic equations of AR with elastic legs through Euler-Lagrange. Due to the complexity of structure, dynamic equations cannot be visibly shown. But, simulation can still be implemented after each loop of simulation by general matrix form of AR. A method of using three P-controllers is introduced to transform MIMO nonlinear underactuated form of AR into a SIMO system. Then, the authors also propose HSM controller, which was well implemented for SIMO system, for motion of AR in one-step period. Although sliding surfaces in that method were proved to work well, a mathematical problem is not completely guaranteed. This led to difficulty in choosing exactly control parameters. GA is proposed to solve that problem. Along with selecting the acceptable control parameters, GA is also successfully used to design the reference trajectories for motion of each link. The success of this approach in control AR robot is consolidated through simulation.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.