Hierarchical Sliding Mode Algorithm for Athlete Robot Walking

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 slidingmode controller is implemented to control basic posture of athlete robot stepping. Successful simulation results show the motion of athlete robot.


Introduction
Two-legged robot is an interesting topic which lasts for long time [1][2][3].One of the classical forms of this type, humanoid robot, which only has solid links, can move or balance through ZMP method [4][5][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][8][9][10][11][12][13] put elastic components for smooth dynamic motion.By these researches [7][8][9][10][11][12][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][5][6] become useless for ARs due to structure of AR as a MIMO underactuated system.Some researches [16][17][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][21][22][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][22][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.

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: Figure 3: Correlation between leg of solid robot and elastic leg of AR.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 as Total potential energy of system is Kinetic energy of system is Lagrange operator is By using Euler-Lagrange method, dynamic equations will be From ( 5), the dynamic model can be written as where , 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 in Also, denote by   and   the reference signal of variables   and   .Consider  0 = const > 0 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][26][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  1 ,  2 ,  3 ,  4 , and Reference trajectories of  4 and  5 are described in Figure 9.
Coordinates (  ,   ) and (  ,   ) have to be selected through GA.

Control Algorithm
In order to use HSC algorithms, new variables will be defined as where The dynamic model ( 6) is rewritten as A simply proportional controller for links 3, 4, and 5 is proposed Also, define where  is the second column of matrix  −1 .
The AR is a five-order system and it is impossible to calculate directly  −1 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 be Equation ( 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   →∞   → 0. 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 chosen where  −1 = const and  0 =  0 = 0.
Derivative (16) with respect to time yields From ( 14) and ( 15), we deduce that the th layer SMC comprises the information of ( − 1)th, ( − 2)th, . . ., 1st layer, and subsystem layer.Hence, define the th layer sliding mode control law as Here,  0 = 0.And  sw ,  eq is switching and equivalent control law for th layer.Let Define Lyapunov function for th layer as Derive (20), and from (17), we obtain Derive ( 14), and substitute ( 14) and ( 18) into new results which is obtained by deriving (14), which yields sw ( = 1, 2, . . ., 5) . ( By considering stability of th layer sliding surface, let From ( 22) and ( 23), switching control law of th layer can be obtained as and the final control law for controller is selected as From 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 th 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 obtain ( From (30), it means that lim →∞   = 0. 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 satisfied and   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: (iii) Half-period time of a motion cycle (time for a step of AR):  (if AR moves fast, then  is chosen in the range: 0 (s) ≤  ≤ 2 (s)) The individuals chosen for crossover process have to satisfy the following conditions: (i) All   ,   , and   ( = 1, 5) are positive for  = 0 → .
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   → 0 and ∑ 5 =1  2  → 0. However, sample time of step  is also chosen through GA.Only ∑ 5 =1  2  cannot represent the ability of tracking trajectory in a period of time.Therefore, fitness function should be selected as  = (1/) ∑ 5 =1  2  .The detected result which generates smaller  from (34) will be considered as better.
The GA process is shown in Figure 12.

Start
Define system parameters and control parameters for simulation Update system variables and Update the control signal

Calculate later variables from equations that describe AR as below
Simulation is required to be ended?

Simulation Results
System parameters are selected as follows: Initial values of variables at  = 0 (s) are specified in Figure 8. Motions of AR under HSM controller are described in Figures 13 and 19.
From initial position (Figure 13(a)), after a period time , AR moves to a new position (Figure 13(b)).An additional time  add = 0.11 (s) 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  +  add .Trajectories of each link variables   are shown in Figures 14-18.
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).

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.

Figure 1 :
Figure 1: Motion of disabled people with elastic legs.

Figure 7 :
Figure 7: Model of AR with variables are referenced to the vertical axis.

Figure 8 :
Figure 8: Position of two legs.(a) Before the step and (b) after the step.

5 Figure 11 :Figure 12 :Figure 13 :Figure 14 :
Figure 11: Hierarchical sliding surfaces structure for system which has one control input and five output variables.

Figure 19 :
Figure 19: Motion description of AR.(a) From 0 s to 0.2 s.(b) From 0.2 s to 0.4 s.(c) From 0.4 s to 0.6 s.(d) From 0.6 s to 0.8 s.(e) From 0.8 s to 1 s.(f) From 1 s to 1.12 s.