1. Introduction

Gait for moving forward or backward or even turning is defined as the alternating phases of single and double support mode of an anthropomorphic robot. The DSP is the phase when both legs are in contact with the navigation surface. In SSP only a single leg is in contact with the ground which is called as the stance leg while the other one is called as the swing leg. The main requirement of the robot is to alternate the phases by maintaining the displacement of the horizontal components such as the Centre of Mass (CoM), with strictly monotonic fashion which means the CoM should be stable without rocking forward and backward. In order to understand the gait, it is necessary to visualize the human planes of sections. The Sagittal Plane divides the body into right and left sections. The Frontal or Coronal Plane separates the body into front and back portions. Finally the Transverse Plane is perpendicular to both the Sagittal and Frontal Planes and divides the body up and down portions at the hip position of the body [1,2]. These three different planes are represented in Figure 1.

Leonardo de Vinci is considered as the first man who has drawn a humanoid mechanism in 1495 [3,4]. Construction and development period of humanoid system commences in the 19^th century when John Brainerd invented the Steam Manin 1865 [4,8]. At the beginning of 20^th century an evolutionary number of humanoid systems appear such as ELEKTRO, BIPER, Tron-Xm, H6, Waseda Legged series, WABOT, WABIAN family, WAP series, SAIKO, E0 to E6, P1 to P3 and so on [4]. Jumping robot, Spring Turkey, Spring Flamingo, Uniroo, Biped Planar, and 3D Biped were the remarkable invention from MIT having noteworthy performance on walking and running movements with dynamic and stable gait. ASIMO the currently most advance humanoid system is demonstrated in the year 2000 [5,6,9]. BIP2000, AIBO, RABBIT, KHR, HUBO, HRP, HOAP, ROBIAN, NAO, iCub, CB2, MAHRU, REEM, QRIO are some of the greatest anthropoid platform came out during the last ten years [4-9].

A type of walking pattern and sensory feedback control strategy were proposed for an anthropomorphic robot stepping up stairs, in 2008 [10]. The parameter selection for the designed gait was excogitated based on the Constrained Nonlinear Optimization Problem (CNOP). Using the Reinforcement Learning (RL) method the control parameters for feedback were adjusted for stepping up the stairs. An algorithm was proposed by S. H. Kim and Y. Sankai to generate extended task for anthropoid to perform various movements including the ascending steps [11]. The third-order Bezier curve was adopted for this experiment where the desired trajectory was possible to accomplish by modifying the control parameters. Accelerometer and Force sensor based control mechanism for a humanoid system stepping up stairs was explicated by T. H. S. Li et al. [12] where an autonomous control strategy was proposed. Self balancing gait using Fuzzy Logic (FL) and Zero Moment Point (ZMP) were adopted for this experiment and ten step stairs climbing strategy was demonstrated with successful accomplishment. Robust model based real time 3D tracking strategy was demonstrated in 2007 where a programmable graphics hardware was used to operate the humanoid robot [13].

Figure 1. Three different Section Planes of an anthropomorphic system

All the methods, mentioned above, have demonstrated with the robustness of their individual gaits having the same type of foot placing strategy as, Flat-Lifting and Flat-Landing (FL-FL) footsteps. This technique is feasible for slow walking patterns. In this paper a novel gait is designed and demonstrated using the BIOLOID humanoid platform where human like walking is demonstrated and can be abridged as, Heel-Contact and Toe-Off (HC-TO) walking strategy. This particular method may provide the ability to the anthropomorphic system for faster walking and even for running. The paper also represents the CoP-CoM tracking strategy which is excogitated by following the ZMP technique and can be defined as, tracking the Centre of Pressure (CoP) movements by CoM of the system. The paper mainly focuses on the gait design strategy and various movement behaviors of the joint actuators in performing the gait for step up-down stairs.

Pre formulation of gait vector and designing the gaits for Ascending-Descending stairs are presented in Section 2. In Section 3, the various angular positions of the joint actuators are tabulated. The deportments of the joint actuators are also plotted and analyzed in the same section. Finally, Section 4 presents the abridgement of the paper.

2. Gait Analysis and Design

2.1. Analysis and Pre Formulation

Humanoid gaits are the combination of various postures and poses. Some series of poses can be accumulated together as a page and a group of pages are combined with an adjustable sequence to make the robot able to achieve the next stage of the performing gait. If the system begins to execute a particular action, the execution of a particular page should be completed before the start of the next action. A mathematical expression for the execution process can be defined with , which is considered as a vector and a function of the joint vector The indicates the pose of the humanoid for a certain moment of time in a particular direction while executing a page. If the start and end poses of a single page is indicated as P_i and P_f, a general equation can be formulated as shown in Equation (1) and Equation (2) where is a set of the joint vectors of ‘i’th pose contingent on the ‘n’ DoF of the system.

(1)

(2)

2.2. Defining the Stair Constraints

Moving up and down stairs are the complex task for a humanoid where more stable gaits are required. During the climbing stairs, the humanoid has to move its CoM position upwards to shift its torso on the next step which is more complex than moving the CoM while navigating on flat surface. Step down stairs is also the difficult and complex chore to remain stable while gravitational force attracts the CoM point. Figure 2 represents the designed stairs having two steps up and down. Height of the each step is about 3.1 cm which makes 6.2 cm height in total for the two steps. The stairs are especially designed for the humanoid with the necessary space to place its feet on the steps.

Figure 2. Two step stairs model

2.3. Gait for Ascending Stairs

As the two swing modes are involved to gait the humanoid for a single step up stairs, the designed gait is conceived with fifteen poses. The first three poses, shown in Figure 3(a) and (b) are considered as same as the walking or obstacle overcoming gait design. The first swing phase for the action leg has to execute five poses to settle the action foot on the step of the stairs as shown in Figure 3(b) to (f). During this action the CoM of the system is shifted backwards for a little distance to make the humanoid stable without falling down, Figure 3(e). With the proper placement of the front foot on the step, the humanoid moves its torso forward in two different poses, Figure 3(g) and (h). Before the beginning of the second swing phase with the rear foot, the anthropoid moves its CoM gradually upwards to fix its stability by front foot support, Figure 3 (i). The second swing phase continues with the next four poses as shown in Figure 3 (j) to (m) and finally the robot moves its torso upwards to come to the Action Pose again. The whole process can be performed in two ways depending on the action leg, right or left. As a result, the algorithm is designed with two functions, StUP_Left( ) and StUP_Right( ). The mathematical description of the procedure can be presented as follows where stands for Step Up vector.

(3)

Here stands for a set of the joint vectors in a certain pose of the system. Figure 3 presents the design of the fifteen poses to perform step up stairs.

Figure 3. Fifteen different poses for a single step up of the stairs, (a) Action Pose and Tilta Pose (two poses), (b) DS-SSa Pose, (c) Foota Lifting Pose, (d) Foota Forward Pose, (e) Foota Adjust Pose, (f) SS-DSa Pose, (g) SS-DSa Complete Pose, (h) Tiltb Pose, (i) CoM Lifting Pose, (j) DS-SSb Pose, (k) Footb Lifting Pose, (l) Footb Forward Pose, (m) SS-DSb Pose and (n) Action Pose

The StUP_Left ( ) and StUP_Right ( ) functions are defined where the series of poses are executed sequentially. To define the main function for the Stair Up Step, StUP ( ) is defined, Figure 4. The function takes a parameter value indicating the left or right leg in action to perform the task. The return value, 0, represents the system as in Action Pose and the completion of the operation successfully.

Figure 4. Defining the Step Up function as StUP ( ) with a parameter value indicating the action leg

Figure 5. Various poses for step up stairs experimented using BIOLOID humanoid

2.4. Fashion of Descending Stairs

Gait for Step down stairs is one of the critical tasks for the humanoid where total of seventeen poses are executed sequentially. After the Action Pose of this gait, the humanoid drops its CoM point downwards by bending both of the knees as shown in Figure 6 (b). After that the robot tilts to one side to start the first swing phase. The swing phase continues with three poses and ends up by touching the down step with the heel of the swing foot as shown in Figure 6 (c) to (g). During the next four poses the system moves its torso forward, upward, forward and finally a little bit backward to ensure the stability transfer from rear leg to front leg. The second swing phase starts after this action as shown in Figure 6 (k). Next five poses are performed one after another to reach at the final stage of the second swing phase, Figure 6 (p). The whole process is completed with the Action Pose of the bipedal system. Based on the general equation as represented in Equation (1) and Equation (2), the strategy is formulated as,

(4)

Here represents the Step Down Stairs vector as the function of a set, , of the joint vectors in a certain pose of the gait. The gait may start by activating any of the left or right leg. The functions, StDN_Left ( ) and StDN_Right ( ) is possible to define where the series of poses are executed by the system in a serial manner. Defining the main function for Step Down Stairs as StDN ( ), as presented in Figure 7.

Figure 6. Seventeen various poses to perform a single step down of the stairs, (a) Action Pose, (b) CoM Drop Pose, (c) Tilta Pose, (d) DS-SSa Pose (e) Foota Forward Pose, (f) Foota Adjust Pose, (g) SS-DSa Pose, (h) SS-DSa Complete Pose, (i) CoM Lifting Pose, (j) CoM Forward Pose, (k) Tiltb Pose, (l) DS-SSb Pose, (m) Footb Forward1 Pose, (n) Footb Forward2 Pose, (o) Footb Adjust Pose (p) SS-DSb Pose and (q) Action Pose

Figure 7. Defining the Step Down function as StDN ( ) with a parameter value indicating the action leg

Figure 8. Various poses for step down stairs applied on BIOLOID humanoid where (p) represents the SS-DSb & Action poses

3. Joint Demeanors and Discussions

The various angular positions of the rotary joint actuators are represented in a tabulated format. All the required angular positions are calculated depending on the various poses of the various gaits where the Geometrical Analysis Technique (GAT) is imposed. The findings of the GAT strategy are experimented on the real time system where a small-scale adjustment is enforced to establish the robustness of the various gaits for the robotic platform. These adjustments are necessary because of the backlash errors of the servo actuators gear heads. Finally the table which is required for each gaits are updated, as shown in Figure 9, and fixed into the robot controller.

To observe the behaviors of the various actuators of the system, the tabulated pose angles of the different gaits are simulated in MatLab using the General Spline Interpolation method as Spline = spapi( k , x , y ) and fnplt( Spline ), where the first argument in spapi( ), k, is the order of the interpolating Spline.

3.1. Analysis of Joint Behaviors

Ascending and Descending stairs are the laborious task for the biped system. In this section, the two separate gaits with different behavior of joint movement patterns are observed for the Up going and the Down going steps. The mirror joint actuator movement characteristics of the performed actions are compared by simulating the tabulated values of the designed gait patterns of the humanoid platform. Figure 10 combines the various trajectories for the Shoulder and Elbow joint actuators where the robot performs the one step up and one step down of the stairs. The rest of the figures from Figure 11 to Figure 16 express the various deflections that occur on the lower torso actuators for the same experiment.

Figure 9. Tabulated values of the corresponding angular positions of the joint actuators for various poses of Ascending and Descending Steps

Figure 10. Upper torso mirror actuator movement patterns to execute the gaits for Step Up and Down Stairs

Figure 11. Motion trajectories of Hip Yaw mirror actuators while performing Up and Down Stairs

Figure 12. Motion trajectories of Hip Roll mirror actuators

Figure 13. Motion trajectories of Hip Pitch mirror actuators

Figure 14. Motion trajectories of Knee Pitch mirror actuators

Figure 15. Motion trajectories of Ankle Pitch mirror actuators for Up and Down Steps of the Stairs

Figure 16. Actuator Motion trajectories for Ankle Roll mirror joints while stepping Up and Down Stairs

The behavior of the Hip Yaw and Ankle Roll joint actuators are showing almost same trajectory paths for their corresponding mirror actuators, indicating in Figure 11 and Figure 16. These characteristics also represents that the z-axis assignment for these joint coordinates are in the same direction. Although the motion trajectories for Hip Roll mirror actuators show a little bit diversion, Figure 12, their z-axis for local frames are also in the same direction. All the other Pitch actuators which are mounted at the lower torso of the robot present the opposite deflection characteristics for their corresponding mirror actuators because of the z-axis assignment of the joint frames are in opposite direction to each other. These characteristics are represented from Figure 13 to Figure 15. Basically these natures of the joint actuators are significant to perform the stable gait for ascending and descending steps of the stairs. The first half or the each graphs represents the pattern for stepping up stairs. The fashion of step down stairs can be expressed by the second half of the graphs, shown from Figure 10 to Figure 16.

4. Conclusions

As the experiment is a part of the main project to establish an optimal navigation system in indoor environment, the robustness of the designed gait is ensured through FK and IK analysis to identify the D-H parameters and imposing GAT or G-T formulation technique. The CoP-CoM tracking strategy is formulated based on the various patterns of the footsteps and zigzag motion of the CoP point where the CoM follows the CoP movement path. The ZMP concept is imposed to have the basic strategy of this analysis. As no sensors are used to acknowledge the environment or contact points on the ground, the CoP point is selected based on the percent of total pressure changes on the straight line between the ankle points of two legs. The percent of pressure is calculated automatically contingent upon a mathematical equation which is excogitated from the predefined step length, step angles and foot lengths. The robot performs its task to step over the stairs comparatively in slow motion but in a robust way by maintaining its torso in upright position. Basically a human like walking gait is tried to design that is experimented using the BIOLOID system and the joint movement behaviors of this robot are analyzed and presented in this paper. The experimental result shows that such kind of gait is feasible to implement on a real size humanoid robot to establish a navigation strategy in indoor environment.

ACKNOWLEDGEMENTS

The authors would like to thank their honorable parents. They also would like to express their gratitude to the Ministry of Higher Education (MOHE), Malaysia, in funding the project through the Fundamental Research Grant Scheme (FRGS).