# An Approach to Stable Walking over Uneven Terrain Using a Reflex-Based Adaptive Gait

Authors: Umar Asif; Javaid Iqbal
Journal: Journal of Control Science and Engineering
 (2011)

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## Abstract

This paper describes the implementation of an adaptive gait in a six-legged walking robot that is capable of generating reactive stepping actions with the same underlying control methodology as an insect for stable walking over uneven terrains. The proposed method of gait generation uses feedback data from onboard sensors to generate an adaptive gait in order to surmount obstacles, gaps and perform stable walking. The paper addresses its implementation through simulations in a visual dynamic simulation environment. Finally the paper draws conclusions about the significance and performance of the proposed gait in terms of tracking errors while navigating in difficult terrains.


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## Body

## 1. Introduction

In the field of motion planning for legged robots, a large body of research work is inspired from insects due to the fact that they are responsive, adaptive and possess sensory systems to generate reactive walking patterns [1]. In the field of biomimetic robots, a large volume of work [2–6] exists which deals with their modeling and design inspired from legged creatures found in nature.It has been well understood through various research studies that mammals and walking insects posses inherent capabilities to choose stable and secure walking patterns in response to external disturbances when the terrain becomes uneven to keep up a continuous gait. This walking pattern is characterized by the sequential motion of legs and coordinated advancement of the body which makes them appropriate and suitable for replicating in walking robots for the real world. A wave gait is the typical gait which is considered to possess intrinsic terrain adaptability. On the other hand, a tripod gait [7] is well known for walking with high mobility due to its fast gait speed and greater static stability.


### 1.1. Literature Review and Related Work

In the field of navigation in difficult terrains, a large volume of research work deals with the development of terrain adaptation control algorithms to realize efficient and stable locomotion [8–10]. Among few works, a motion planner based on a distributed neural network controller [11], gait generation methods using energy-based stability margins to understand reliable locomotion [12], fault detection schemes [13], development of fault tolerant cyclic gaits [14], and gait planning based upon kinematic control of hexapod robots [15] are some investigations with excellent results.Earlier robot designs such as [16] focused on the development of behavior based strategies to formulate the robot-environment interactions in order to design adaptive walking behaviors using sensory information. Later on, studies such as [17–19] investigated the design of robotic systems with their prime objective to realize the walking behaviors of insects in real walking machines. Thus, the objective of these research studies has always been an effort to devise schemes to define leg movements which replicate those of actual insects in order to realize biologically inspired walking. Curse worked on such gait generation methodologies by investigating the walking behaviors of stick insects [20, 21]. An appreciable contribution in this context was made by the researchers Hess and Büschges [22] and Ekeberg et al. [1], who studied the insect neurobiology and investigated that the stick insects control their leg movements through sensory oscillators in their joints and thoracic ganglia. Therefore, their joint movements in a particular leg are influenced by sensory signals from the brain as well as from the joints in other legs to generate reactive stepping patterns. Since, the patterns are reactive the resulting locomotion is responsive and adaptive that allows the insect to conform to its underlying terrain. Such neurobiological-based gait generation methodologies have been realized using computer simulations [1] with one and two legged robots [23]. A closely related study [24] deals with an excellent implementation of such a neurobiological leg control system identified in the thoracic ganglia of stick insects using a hexapod robot by modeling biologically plausible sensory pathways for pattern generation. The work [24] describes these pathways using advanced algorithm like “Elevator Reflex” to surmount obstacles and “Searching Reflex” to surmount gaps and holes in order to improve navigation over rough terrains with excellent results.While these research investigations are sufficient to demonstrate the success of adaptive locomotion, most of the work deals with implementation on either smooth flat surfaces or structured environments only. In natural terrains, the formation of impulsive forces during the impact (robot-ground interaction) giving rise to disturbances, affects the walking gait and offers a great challenge to realize locomotion with adequate stability and performance. Therefore, in contrast, this paper aims to describe a biologically inspired gait generation method for walking over uneven terrains with an ability to traverse through obstacles and gaps with minimal tracking errors.The first part of the paper describes our six-legged robot in terms of its modeling and design. The second part of the paper describes our proposed foothold reflex control method to generate a reactive and adaptive gait, able to plan footholds and appropriate leg sequences to change support points and transverse a rough terrain.


## 1.1. Literature Review and Related Work

In the field of navigation in difficult terrains, a large volume of research work deals with the development of terrain adaptation control algorithms to realize efficient and stable locomotion [8–10]. Among few works, a motion planner based on a distributed neural network controller [11], gait generation methods using energy-based stability margins to understand reliable locomotion [12], fault detection schemes [13], development of fault tolerant cyclic gaits [14], and gait planning based upon kinematic control of hexapod robots [15] are some investigations with excellent results.Earlier robot designs such as [16] focused on the development of behavior based strategies to formulate the robot-environment interactions in order to design adaptive walking behaviors using sensory information. Later on, studies such as [17–19] investigated the design of robotic systems with their prime objective to realize the walking behaviors of insects in real walking machines. Thus, the objective of these research studies has always been an effort to devise schemes to define leg movements which replicate those of actual insects in order to realize biologically inspired walking. Curse worked on such gait generation methodologies by investigating the walking behaviors of stick insects [20, 21]. An appreciable contribution in this context was made by the researchers Hess and Büschges [22] and Ekeberg et al. [1], who studied the insect neurobiology and investigated that the stick insects control their leg movements through sensory oscillators in their joints and thoracic ganglia. Therefore, their joint movements in a particular leg are influenced by sensory signals from the brain as well as from the joints in other legs to generate reactive stepping patterns. Since, the patterns are reactive the resulting locomotion is responsive and adaptive that allows the insect to conform to its underlying terrain. Such neurobiological-based gait generation methodologies have been realized using computer simulations [1] with one and two legged robots [23]. A closely related study [24] deals with an excellent implementation of such a neurobiological leg control system identified in the thoracic ganglia of stick insects using a hexapod robot by modeling biologically plausible sensory pathways for pattern generation. The work [24] describes these pathways using advanced algorithm like “Elevator Reflex” to surmount obstacles and “Searching Reflex” to surmount gaps and holes in order to improve navigation over rough terrains with excellent results.While these research investigations are sufficient to demonstrate the success of adaptive locomotion, most of the work deals with implementation on either smooth flat surfaces or structured environments only. In natural terrains, the formation of impulsive forces during the impact (robot-ground interaction) giving rise to disturbances, affects the walking gait and offers a great challenge to realize locomotion with adequate stability and performance. Therefore, in contrast, this paper aims to describe a biologically inspired gait generation method for walking over uneven terrains with an ability to traverse through obstacles and gaps with minimal tracking errors.The first part of the paper describes our six-legged robot in terms of its modeling and design. The second part of the paper describes our proposed foothold reflex control method to generate a reactive and adaptive gait, able to plan footholds and appropriate leg sequences to change support points and transverse a rough terrain.


## 2. Robot Modeling and Design

### 2.1. Physical Model

The mechanical structure of our robot as shown in Figure1, consists of six identical legs placed around the main body in a biologically inspired configuration to enable omni-directionability and enhance maneuverability. In order to describe the kinematic model of the robot leg inspired from a stick insect, the work presented in [25] has been reviewed. There are four functional segments of the leg of a stick insect: the coxa (c), the femur (f), the tibia (t), and the foot. For simplicity, a stick insect leg can be modelled as a serial manipulator with three hinge joints by neglecting the joint of the foot and the foot itself, resulting in three degrees of freedom as investigated in [25]. Thus, the leg of our robot consists of three links namely: coxa, femur and tibia, interconnected through three revolute joints. From the body towards the foot, the joints are called body-coxa joint (θc), coxa-femur joint (θf), and femur-tibia joint (θt). Figure 2 portraits the kinematic configuration of the leg in terms of its links and their motion envelope.Figure 1
Pro engineer model of hexapod robot.Figure 2
Structure of an individual leg.


### 2.2. Reachable Workspace

If all the joints of a leg return appropriate or desirable angles for a reference pose in a given space, then that pose is considered to be a subset of manipulator’s working space. In this context, analytic and diagrammatizing methods are typically used to work out the robot’s working space. On the other hand, the numerical methods have advantages such as fast speed, high precision, easy operation, large application range, and adaptability to various types of robotic structures. As a consequence, it is widely used by researchers as investigated in [26, 28].Figure3 shows the working envelopes of all the legs attached to the main body as a reachable workspace represented by solid annulus regions. Each leg has a reachable area in the form of a sector as defined in [29], where adjacent or neighboring areas cause overlap of each other’s region. Thus, for avoiding the interference problem, a dexterous workspace is determined contained inside the reachable workspace of each leg as shown in the Figures 2 and 3. The dexterous workspace is an area on the plane mapped on the terrain within which a leg can be placed effectively anywhere while satisfying the static stability.Figure 3
Working envelopes of robot legs.


### 2.3. Kinematic Model

The kinematic model of our robot consists of forward and inverse kinematic equations formulated using Denavit-Hartenberg convention as investigated in [30], further described by (1).(1)FootxiG=cos⁡⁡γcos⁡⁡φPxftiB-cos⁡⁡γsin⁡⁡φPyftiB+sin⁡⁡γPzftiB,FootyiG=(cos⁡⁡ψsin⁡⁡φ+sin⁡⁡ψsin⁡⁡γcos⁡⁡φ)PxftiB+(cos⁡⁡ψcos⁡⁡φ-sin⁡⁡ψsin⁡⁡γsin⁡⁡φ)PyftiB-sin⁡⁡ψcos⁡⁡γPzftiB,FootziG=(sin⁡⁡ψsin⁡⁡φ-cos⁡⁡ψsin⁡⁡γcos⁡⁡φ)PxftiB+(sin⁡⁡ψcos⁡⁡φ+cos⁡⁡ψsin⁡⁡γsin⁡⁡φ)PyftiB+cos⁡⁡ψcos⁡⁡γPzftiB,θci=ATAN2(FootziG,FootxiG),θfi=ATAN2(ti,-FootyiG)+ATAN2((-FootyiG2+ti2-(ti2+lf2+FootyiG2-lt2)2×lf2),(ti2+lf2+FootyiG2-lt2)2×lf),θti=ATAN2((-FootyiGcos⁡⁡θf-sin⁡⁡θfti),(ticos⁡⁡θf-lf-sin⁡⁡θfFootyiG)),ti=FootxiGcos⁡θc-lc+FootziGsin⁡θc,CoMiG=∑mi×pCoMiG∑mi,xiZMP=∑i=1nmi(xi(ÿi+gy)-xïyi)∑i=1nmixi(ÿi+gy),ziZMP=∑i=1nmi(zi(ÿi+gy)-zïyi)∑i=1nmizi(ÿi+gy),mi=massofthelinki;xi,yi,zi=positionvectoroflinki;ẍi,ÿi,z̈i=accelerationsoflinki;gy=gravityvector;CoMiG=centerofmassoflinki;[xiZMPziZMP]=zeromomentpointpositionvector;[FootxiGFootyiGFootziG]=footholdpositionvector;[θci,θfi,θti]=q̅i=jointsrotationangles. In order to ensure the performance of a walking gait, the performance measures are typically evaluated in terms of gait stability margins using zero moment point (ZMP) based stability criterion. Therefore, it is necessary to consider and define the ZMP-based stability criterion. The definition of ZMP adopted in this paper is as described in [31, 32], which is: ZMP is the projected point of the resultant forces of the gravity and the inertia forces acting on the robot to the ground, to which the moments of the resultant forces equal to zero. To ensure dynamic stability, the ZMP must lie within the Support Polygon on Surface (SPOS) at all the times and does not lie on the edges of the SPOS. SPOS is the convex hull (polygon) of the footholds in contact with the ground. If the ZMP reaches the edge of the SPOS, the walking gait will develop a high tendency to become unstable and may result in tumbling of the robot on the ground. The position of ZMP is derived from [31, 32], as described in (1).


## 2.1. Physical Model

The mechanical structure of our robot as shown in Figure1, consists of six identical legs placed around the main body in a biologically inspired configuration to enable omni-directionability and enhance maneuverability. In order to describe the kinematic model of the robot leg inspired from a stick insect, the work presented in [25] has been reviewed. There are four functional segments of the leg of a stick insect: the coxa (c), the femur (f), the tibia (t), and the foot. For simplicity, a stick insect leg can be modelled as a serial manipulator with three hinge joints by neglecting the joint of the foot and the foot itself, resulting in three degrees of freedom as investigated in [25]. Thus, the leg of our robot consists of three links namely: coxa, femur and tibia, interconnected through three revolute joints. From the body towards the foot, the joints are called body-coxa joint (θc), coxa-femur joint (θf), and femur-tibia joint (θt). Figure 2 portraits the kinematic configuration of the leg in terms of its links and their motion envelope.Figure 1
Pro engineer model of hexapod robot.Figure 2
Structure of an individual leg.


## 2.2. Reachable Workspace

If all the joints of a leg return appropriate or desirable angles for a reference pose in a given space, then that pose is considered to be a subset of manipulator’s working space. In this context, analytic and diagrammatizing methods are typically used to work out the robot’s working space. On the other hand, the numerical methods have advantages such as fast speed, high precision, easy operation, large application range, and adaptability to various types of robotic structures. As a consequence, it is widely used by researchers as investigated in [26, 28].Figure3 shows the working envelopes of all the legs attached to the main body as a reachable workspace represented by solid annulus regions. Each leg has a reachable area in the form of a sector as defined in [29], where adjacent or neighboring areas cause overlap of each other’s region. Thus, for avoiding the interference problem, a dexterous workspace is determined contained inside the reachable workspace of each leg as shown in the Figures 2 and 3. The dexterous workspace is an area on the plane mapped on the terrain within which a leg can be placed effectively anywhere while satisfying the static stability.Figure 3
Working envelopes of robot legs.


## 2.3. Kinematic Model

The kinematic model of our robot consists of forward and inverse kinematic equations formulated using Denavit-Hartenberg convention as investigated in [30], further described by (1).(1)FootxiG=cos⁡⁡γcos⁡⁡φPxftiB-cos⁡⁡γsin⁡⁡φPyftiB+sin⁡⁡γPzftiB,FootyiG=(cos⁡⁡ψsin⁡⁡φ+sin⁡⁡ψsin⁡⁡γcos⁡⁡φ)PxftiB+(cos⁡⁡ψcos⁡⁡φ-sin⁡⁡ψsin⁡⁡γsin⁡⁡φ)PyftiB-sin⁡⁡ψcos⁡⁡γPzftiB,FootziG=(sin⁡⁡ψsin⁡⁡φ-cos⁡⁡ψsin⁡⁡γcos⁡⁡φ)PxftiB+(sin⁡⁡ψcos⁡⁡φ+cos⁡⁡ψsin⁡⁡γsin⁡⁡φ)PyftiB+cos⁡⁡ψcos⁡⁡γPzftiB,θci=ATAN2(FootziG,FootxiG),θfi=ATAN2(ti,-FootyiG)+ATAN2((-FootyiG2+ti2-(ti2+lf2+FootyiG2-lt2)2×lf2),(ti2+lf2+FootyiG2-lt2)2×lf),θti=ATAN2((-FootyiGcos⁡⁡θf-sin⁡⁡θfti),(ticos⁡⁡θf-lf-sin⁡⁡θfFootyiG)),ti=FootxiGcos⁡θc-lc+FootziGsin⁡θc,CoMiG=∑mi×pCoMiG∑mi,xiZMP=∑i=1nmi(xi(ÿi+gy)-xïyi)∑i=1nmixi(ÿi+gy),ziZMP=∑i=1nmi(zi(ÿi+gy)-zïyi)∑i=1nmizi(ÿi+gy),mi=massofthelinki;xi,yi,zi=positionvectoroflinki;ẍi,ÿi,z̈i=accelerationsoflinki;gy=gravityvector;CoMiG=centerofmassoflinki;[xiZMPziZMP]=zeromomentpointpositionvector;[FootxiGFootyiGFootziG]=footholdpositionvector;[θci,θfi,θti]=q̅i=jointsrotationangles. In order to ensure the performance of a walking gait, the performance measures are typically evaluated in terms of gait stability margins using zero moment point (ZMP) based stability criterion. Therefore, it is necessary to consider and define the ZMP-based stability criterion. The definition of ZMP adopted in this paper is as described in [31, 32], which is: ZMP is the projected point of the resultant forces of the gravity and the inertia forces acting on the robot to the ground, to which the moments of the resultant forces equal to zero. To ensure dynamic stability, the ZMP must lie within the Support Polygon on Surface (SPOS) at all the times and does not lie on the edges of the SPOS. SPOS is the convex hull (polygon) of the footholds in contact with the ground. If the ZMP reaches the edge of the SPOS, the walking gait will develop a high tendency to become unstable and may result in tumbling of the robot on the ground. The position of ZMP is derived from [31, 32], as described in (1).


## 3. Terrain Adaptation

Terrain exploration is considered to be an important aspect in motion planning of walking robots that has been previously discussed in more detail in [33–40]. Once a terrain map is available, a foothold selection method can be implemented. In this paper, the terrain knowledge is supplied to the robot offline in form of terrain’s latitude, longitude, and altitude information. The objective is to follow a reference path on a known geographical location using its latitude, longitude, and altitude information. During walking, the robot uses sensor-based traction control to perform straight line walking.


### 3.1. Gait Generation

The gait modeled here constitutes three phases: swing, drop, and stance as described in (2). Each phase is characterized by its unique combination of leg and joint motions coordination. Swing is identified by protraction of body-coxa joint, levation of the coxa-femur joint, and extension of the femur-tibia joint. This combination serves to lift the foothold from FootyiG to Footyi+1G and sway it in the direction of motion to acquire the next foothold position (Footxi+1G,Footzi+1G) in air. Drop phase lowers the leg until contact with ground is verified from foothold’s touch sensor notification. During stance phase, the involved legs both support and propel the body in the direction of desired motion. Joint trajectories (q̅i,q̇̅i,q̈̅i) are computed using linear splines with parabolic blends as described in (2).(2)Swing=f(q̅i)(leglift,legstroke),0<leglift≤15cm,0<legstroke≤10cm,Stance=f(q̅i)(dutyfactor(β),legstroke),leglift=FootyiG→ΔFootyi+1G,legstroke={FootxiG→ΔFootxi+1G,FootziG→ΔFootzi+1G,GaitSpeed=Si=δi+1-δiti+1-ti,β=tstancetcycle⁡,(βωswing+βωstance=1),linearspline{q̅i=q̅i+1+(q̅̇i+2-q̅̇i+1)ti,q̅̇i=q̅̇i+2-q̅̇i+1,q̅̈i=0,parabolicblend{q̅i=q̅i+1+(q̅̇i+2-q̅̇i+1)(ti-tblend)+12q̅̈i+2tblend2q̅̇i=q̅̇i+2-q̅̇i+1+q̅̈i+2tblendq̅̈i=q̅̈i+2tblend=ti-(12ti+1+(ti+2-ti+1))ωswing=swingfrequency;ωstance=stancefrequency;tcycle⁡=totalcycle⁡time;tstance=stancetime;δi=distancetravelled.As described earlier, a leg sensory network is established for each leg that defines a reflex system to inform the main controller that a disturbance requiring a behavior other than normal walking has been encountered. Upon perceiving such an abnormal behavior, the controller generates sensory information based upon disturbance estimation during the impact (leg-environment interaction) that alters the normal sequential motion of joints to adaptive and emergent. This reflex-based gait generation method is termed as foothold reflex control (FRC), further explained in the following Section3.2.


### 3.2. Foothold Reflex Control (FRC)

The Foothold Reflex Control (FRC) method aims to step over surmountable obstacles, sizeable gaps and holes using online traction control. The method makes an estimate of the terrain by using the foot contact point information from the leg sensory network. If all the contact points are at the same height, the robot is considered to be walking on a flat terrain. In such a case, the robot uses a shortleglift and a large legstroke to increase its travel speed.Upon detecting a step or a surmountable obstacle using the sensor information, the FRC initiates the behavior other than normal walking that alters the motions of the involved leg to retraction to a safe zone (an attempt to minimize the risk of snagging on the obstacle), levation in air (an effort to acquire a position higher than the perceived height of the obstacle), and swing to its next foothold planned within the dexterous workspace of the involved leg. In parallel, FRC uses an independent attitude controller to keep the robot’s body leveled above the terrain with online adjustments while climbing and traversing over the gaps using a sensor-based traction control. This phenomenon is further illustrated in Figure4 using dynamic simulation results in order to validate the success of the proposed method.Figure 4
Climbing a step (top) and crossing gaps (bottom) scenarios to present the foothold planning using the proposed Foothold Reflex Control method. Normal walking behavior is momentarily changed upon sensing an obstruction. Once the obstruction is removed (e.g., loss of contact with the obstruction), the normal motions are resumed.Foot interaction with the ground is modeled here based on a general coulomb friction model [41, 42], further described in (3).(3)[Footxi+1GFootzi+1G]=[FootxiGFootziG]+[∬ii+1fxi+miẍimi∬ii+1fzi+miz̈imi],Fti≤μNi,fxi2+fzi2-μNi<0,
where Fti is the total tangential force between the contact point and the terrain, μ is the combined coefficient of ground static and dynamic friction, and Ni is the total normal reaction force to prevent the penetration of the contact point. The coefficients of static and dynamic friction are taken as a combined coefficient of ground friction for simplicity, calculated online using the tangential and normal reaction forces estimated using leg sensors data. The overall control framework is further illustrated in Figure 5.Figure 5
Overall control system framework.


### 3.3. Gait Performance Measures

To be able to generate an optimal gait, it is necessary to determine the performance of the proposed gait generation. In this paper, the performance of the gait is chosen to be measured on the traction of the robot and its stability. As described earlier, the stability of the robot is determined online by evaluating the position of the CoM and ZMP relative to the SPOS.


## 3.1. Gait Generation

The gait modeled here constitutes three phases: swing, drop, and stance as described in (2). Each phase is characterized by its unique combination of leg and joint motions coordination. Swing is identified by protraction of body-coxa joint, levation of the coxa-femur joint, and extension of the femur-tibia joint. This combination serves to lift the foothold from FootyiG to Footyi+1G and sway it in the direction of motion to acquire the next foothold position (Footxi+1G,Footzi+1G) in air. Drop phase lowers the leg until contact with ground is verified from foothold’s touch sensor notification. During stance phase, the involved legs both support and propel the body in the direction of desired motion. Joint trajectories (q̅i,q̇̅i,q̈̅i) are computed using linear splines with parabolic blends as described in (2).(2)Swing=f(q̅i)(leglift,legstroke),0<leglift≤15cm,0<legstroke≤10cm,Stance=f(q̅i)(dutyfactor(β),legstroke),leglift=FootyiG→ΔFootyi+1G,legstroke={FootxiG→ΔFootxi+1G,FootziG→ΔFootzi+1G,GaitSpeed=Si=δi+1-δiti+1-ti,β=tstancetcycle⁡,(βωswing+βωstance=1),linearspline{q̅i=q̅i+1+(q̅̇i+2-q̅̇i+1)ti,q̅̇i=q̅̇i+2-q̅̇i+1,q̅̈i=0,parabolicblend{q̅i=q̅i+1+(q̅̇i+2-q̅̇i+1)(ti-tblend)+12q̅̈i+2tblend2q̅̇i=q̅̇i+2-q̅̇i+1+q̅̈i+2tblendq̅̈i=q̅̈i+2tblend=ti-(12ti+1+(ti+2-ti+1))ωswing=swingfrequency;ωstance=stancefrequency;tcycle⁡=totalcycle⁡time;tstance=stancetime;δi=distancetravelled.As described earlier, a leg sensory network is established for each leg that defines a reflex system to inform the main controller that a disturbance requiring a behavior other than normal walking has been encountered. Upon perceiving such an abnormal behavior, the controller generates sensory information based upon disturbance estimation during the impact (leg-environment interaction) that alters the normal sequential motion of joints to adaptive and emergent. This reflex-based gait generation method is termed as foothold reflex control (FRC), further explained in the following Section3.2.


## 3.2. Foothold Reflex Control (FRC)

The Foothold Reflex Control (FRC) method aims to step over surmountable obstacles, sizeable gaps and holes using online traction control. The method makes an estimate of the terrain by using the foot contact point information from the leg sensory network. If all the contact points are at the same height, the robot is considered to be walking on a flat terrain. In such a case, the robot uses a shortleglift and a large legstroke to increase its travel speed.Upon detecting a step or a surmountable obstacle using the sensor information, the FRC initiates the behavior other than normal walking that alters the motions of the involved leg to retraction to a safe zone (an attempt to minimize the risk of snagging on the obstacle), levation in air (an effort to acquire a position higher than the perceived height of the obstacle), and swing to its next foothold planned within the dexterous workspace of the involved leg. In parallel, FRC uses an independent attitude controller to keep the robot’s body leveled above the terrain with online adjustments while climbing and traversing over the gaps using a sensor-based traction control. This phenomenon is further illustrated in Figure4 using dynamic simulation results in order to validate the success of the proposed method.Figure 4
Climbing a step (top) and crossing gaps (bottom) scenarios to present the foothold planning using the proposed Foothold Reflex Control method. Normal walking behavior is momentarily changed upon sensing an obstruction. Once the obstruction is removed (e.g., loss of contact with the obstruction), the normal motions are resumed.Foot interaction with the ground is modeled here based on a general coulomb friction model [41, 42], further described in (3).(3)[Footxi+1GFootzi+1G]=[FootxiGFootziG]+[∬ii+1fxi+miẍimi∬ii+1fzi+miz̈imi],Fti≤μNi,fxi2+fzi2-μNi<0,
where Fti is the total tangential force between the contact point and the terrain, μ is the combined coefficient of ground static and dynamic friction, and Ni is the total normal reaction force to prevent the penetration of the contact point. The coefficients of static and dynamic friction are taken as a combined coefficient of ground friction for simplicity, calculated online using the tangential and normal reaction forces estimated using leg sensors data. The overall control framework is further illustrated in Figure 5.Figure 5
Overall control system framework.


## 3.3. Gait Performance Measures

To be able to generate an optimal gait, it is necessary to determine the performance of the proposed gait generation. In this paper, the performance of the gait is chosen to be measured on the traction of the robot and its stability. As described earlier, the stability of the robot is determined online by evaluating the position of the CoM and ZMP relative to the SPOS.


## 4. Simulation Results

Initial simulation tests were performed to realize the success of the FRC method to traverse through steps and gaps as illustrated in Figure4. Pictorial representation of the robot climbing and descending steps is further shown in Figures 6 and 7 while traversing over gaps is pictorially illustrated in Figure 8.Figure 6
Pictorial representation of robot climbing steps of variable heights using Foothold Reflex Control (FRC) method.Figure 7
Pictorial representation of robot descending steps of variable depths using Foothold Reflex Control (FRC) method.Figure 8
Pictorial representation of robot traversing through gaps of variable gap lengths using Foothold Reflex Control (FRC) method.


### 4.1. Obstacle Crossing Testing

During these tests, the robot was able to walk with a maximum speed of 5 cm/sec. The robot started with control inputs (leg-lift: 9 cm and leg-stroke: 5 cm) and easily surmounted the initial steps (step heights: 3 cm, 5 cm, 2 cm, 6 cm, 7 cm) without much modification to the planned control variables. Upon perceiving obstruction (step height: 10 cm), the leg retracted 2 cm, levated 6 cm, and extended with a leg-stroke of 7 cm as a consequence of the Foothold Reflex Control method intervention. As this leg-lift of 6 cm was still insufficient to clear the obstacle, the contact switch again detected the surface and the FRC sequential loop re-executed. The robot surpassed the obstruction eventually with a leg-lift of 12 cm and leg-stroke of 7 cm. Similarly, upon detecting obstruction (step height: 15 cm), the leg was elevated 20 cm which was the maximum achievable leg-lift within its dexterous workspace to surmount obstacle. The robot successfully steered through the rest of the descending steps using the same methodology. Raised surfaces of variable widths and variable heights were placed over an uneven terrain perpendicular to the robot’s path of motion in order to test the ability of the controller in navigating through gaps and holes. The first gap was approximately 5 cm deep and 7 cm across, while the second gap was 7 cm deep and 9 cm across. Success was observed in most of the simulation trials.


### 4.2. Walking over Uneven Terrain Testing

In order to evaluate the performance of the proposed FRC method for navigation in irregular terrains, a terrain is setup using dynamic simulation environment provided by Microsoft Robotics Developer Studio. The terrain constitutes an uneven surface-mesh created using heightfield maps to model elevations and depressions which can offer significant disturbances while walking, as shown in Figure9. This terrain offers a big challenge in comparison to flat surfaces in terms of stable locomotion because the elevations and depressions subject the robot to a large deal of disturbances in both lateral and angular directions.Figure 9
Terrain created using heightfield maps in a dynamic simulation environment.In the first simulation test, the FRC method was not used. Figure11 shows the locomotion of the robot in terms of its center of mass (CoM) and zero moment point (ZMP) profiles with the support polygons on surface. As apparent from the figures, when the FRC method was not used, the robot tends to diverge from the straight line paths between the desired waypoints, further illustrated pictorially in Figure 10. The heading and displacement errors are shown in Figure 12, where positive error corresponds to a divergence along the left side of the reference track while the negative portion corresponds to a divergence along the right side of the reference track. The results show a maximum translational error of ±52 cm and a maximum rotational error of ±3 degrees approximately. The terrain negotiated by the robot during walking is shown in Figure 13 as a surface mesh generated using the contact point information. The color bar in Figure 13 represents the height of the terrain along the y axis.Figure 10
Pictorial representation of the path traced by the robot without using the FRC method.Plot of CoM and ZMP with SPOS without using the Foothold Reflex Control method.
(a)(b)Tracking errors of robot locomotion without using the FRC method.
(a)(b)Figure 13
Terrain intercepted by the robot without using the FRC method.A total of 25 simulation tests were conducted using the proposed FRC method. The simulation results are plotted in Figure15 which represents the path traced by the robot. As apparent from Figure 15, both the CoM and ZMP profiles lie inside the Support Polygons on Surface satisfying the dynamic stability constraints and the path traced by the robot closely matches with the reference track as shown pictorially in Figure 14. The terrain explored by the robot is shown in Figure 16.Figure 14
Pictorial representation of the path traced by the robot using the proposed FRC method.Plot of CoM and ZMP with SPOS using the proposed Foothold Reflex Control method.
(a)(b)Figure 16
Terrain intercepted by the robot using the FRC method.Figures17, 18, and 19 demonstrate a comparative evaluation of the locomotion of the robot with and without using the proposed foothold reflex control method represented by blue and red plots respectively. Figure 17 shows a comparative representation of the estimated ground reaction forces. A close inspection of Figure 17(a) reveals that the total tangential force remains lesser than the normal reaction force times the combined coefficient of ground friction which validates the satisfaction of stability criterion throughout the robot’s locomotion, further verified by the blue plot in Figure 18(a). Further evaluation of the simulation results reveals that the disturbances have significantly reduced when using the FRC method as shown in Figure 19. The attitude controller efficiently tracked the robot posture while walking as shown in Figure 21. Average speed of the robot was reduced from 0.33 m/sec to 0.32 m/sec using the proposed reflex control method however, the robot completed the course in relatively lesser time (3.5 mins as compared to 4 mins when not using the FRC method) with minimal tracking errors. The results reported in Figures 19 and 20 are considered satisfactory given the quality of the kinematic model and the precision of software sensors provided by the open dynamics engine.A Representation of ground forces in the two simulation tests.
(a)(b)Plots of No-Slip condition in the two simulation tests.
(a)(b)A comparative representation of disturbance estimation with and without using the proposed FRC method.
(a)(b)(c)Tracking Errors using the proposed FRC method.
(a)(b)Figure 21
Locomotion of robot in visual simulation environment to show the body attitude control.


## 4.1. Obstacle Crossing Testing

During these tests, the robot was able to walk with a maximum speed of 5 cm/sec. The robot started with control inputs (leg-lift: 9 cm and leg-stroke: 5 cm) and easily surmounted the initial steps (step heights: 3 cm, 5 cm, 2 cm, 6 cm, 7 cm) without much modification to the planned control variables. Upon perceiving obstruction (step height: 10 cm), the leg retracted 2 cm, levated 6 cm, and extended with a leg-stroke of 7 cm as a consequence of the Foothold Reflex Control method intervention. As this leg-lift of 6 cm was still insufficient to clear the obstacle, the contact switch again detected the surface and the FRC sequential loop re-executed. The robot surpassed the obstruction eventually with a leg-lift of 12 cm and leg-stroke of 7 cm. Similarly, upon detecting obstruction (step height: 15 cm), the leg was elevated 20 cm which was the maximum achievable leg-lift within its dexterous workspace to surmount obstacle. The robot successfully steered through the rest of the descending steps using the same methodology. Raised surfaces of variable widths and variable heights were placed over an uneven terrain perpendicular to the robot’s path of motion in order to test the ability of the controller in navigating through gaps and holes. The first gap was approximately 5 cm deep and 7 cm across, while the second gap was 7 cm deep and 9 cm across. Success was observed in most of the simulation trials.


## 4.2. Walking over Uneven Terrain Testing

In order to evaluate the performance of the proposed FRC method for navigation in irregular terrains, a terrain is setup using dynamic simulation environment provided by Microsoft Robotics Developer Studio. The terrain constitutes an uneven surface-mesh created using heightfield maps to model elevations and depressions which can offer significant disturbances while walking, as shown in Figure9. This terrain offers a big challenge in comparison to flat surfaces in terms of stable locomotion because the elevations and depressions subject the robot to a large deal of disturbances in both lateral and angular directions.Figure 9
Terrain created using heightfield maps in a dynamic simulation environment.In the first simulation test, the FRC method was not used. Figure11 shows the locomotion of the robot in terms of its center of mass (CoM) and zero moment point (ZMP) profiles with the support polygons on surface. As apparent from the figures, when the FRC method was not used, the robot tends to diverge from the straight line paths between the desired waypoints, further illustrated pictorially in Figure 10. The heading and displacement errors are shown in Figure 12, where positive error corresponds to a divergence along the left side of the reference track while the negative portion corresponds to a divergence along the right side of the reference track. The results show a maximum translational error of ±52 cm and a maximum rotational error of ±3 degrees approximately. The terrain negotiated by the robot during walking is shown in Figure 13 as a surface mesh generated using the contact point information. The color bar in Figure 13 represents the height of the terrain along the y axis.Figure 10
Pictorial representation of the path traced by the robot without using the FRC method.Plot of CoM and ZMP with SPOS without using the Foothold Reflex Control method.
(a)(b)Tracking errors of robot locomotion without using the FRC method.
(a)(b)Figure 13
Terrain intercepted by the robot without using the FRC method.A total of 25 simulation tests were conducted using the proposed FRC method. The simulation results are plotted in Figure15 which represents the path traced by the robot. As apparent from Figure 15, both the CoM and ZMP profiles lie inside the Support Polygons on Surface satisfying the dynamic stability constraints and the path traced by the robot closely matches with the reference track as shown pictorially in Figure 14. The terrain explored by the robot is shown in Figure 16.Figure 14
Pictorial representation of the path traced by the robot using the proposed FRC method.Plot of CoM and ZMP with SPOS using the proposed Foothold Reflex Control method.
(a)(b)Figure 16
Terrain intercepted by the robot using the FRC method.Figures17, 18, and 19 demonstrate a comparative evaluation of the locomotion of the robot with and without using the proposed foothold reflex control method represented by blue and red plots respectively. Figure 17 shows a comparative representation of the estimated ground reaction forces. A close inspection of Figure 17(a) reveals that the total tangential force remains lesser than the normal reaction force times the combined coefficient of ground friction which validates the satisfaction of stability criterion throughout the robot’s locomotion, further verified by the blue plot in Figure 18(a). Further evaluation of the simulation results reveals that the disturbances have significantly reduced when using the FRC method as shown in Figure 19. The attitude controller efficiently tracked the robot posture while walking as shown in Figure 21. Average speed of the robot was reduced from 0.33 m/sec to 0.32 m/sec using the proposed reflex control method however, the robot completed the course in relatively lesser time (3.5 mins as compared to 4 mins when not using the FRC method) with minimal tracking errors. The results reported in Figures 19 and 20 are considered satisfactory given the quality of the kinematic model and the precision of software sensors provided by the open dynamics engine.A Representation of ground forces in the two simulation tests.
(a)(b)Plots of No-Slip condition in the two simulation tests.
(a)(b)A comparative representation of disturbance estimation with and without using the proposed FRC method.
(a)(b)(c)Tracking Errors using the proposed FRC method.
(a)(b)Figure 21
Locomotion of robot in visual simulation environment to show the body attitude control.


## 5. Conclusion

The robot presented here is a hexapod to walk using a biologically inspired leg control system over an uneven terrain using a sensor-based traction control. Sensory network in a leg is responsible for influencing the direction and motion of each joint such that an evolving, adaptive, and reactive stepping pattern is realized through leg-environment interaction. Simulation tests that evaluated this gait model showed that the robot could clear raised and lowered obstructions over uneven surface which were within and beyond the nominal gait control variables (leg-lift and leg-stroke), as shown by the obstacle crossing testing results. The simulation testing of the proposed foothold reflex control method showed significant improvement in the robot’s ability to track a reference path over an uneven terrain. The gait stability was ensured using the center of mass and zero moment point stability criterion. The novelty of the proposed navigation method thus lies in the successful implementation of a reflex-based gait generation with an ability to perform obstacle crossing and path following in irregular terrains with adequate stability and minimal tracking errors. Future work in this area will involve improving fusion of sensor data through advanced filtering and disturbance rejection techniques.



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*Source: 783741-2011-09-28.xml*