Gait Generation and Mechatronic Design of Planar Walker .... max l. DEFINITION. 1: The state of actuators in the Planar Walker is an 8-tuple. , where. 1. 2. 3.
Gait Generation and Mechatronic Design of Planar Walker I-Ming Chen and Song Huat Yeo School of Mechanical and Production Engineering, Nanyang Technological University, Singapore
Abstract. A surface walking/climbing robot is developed based on a simple planar 8-bar locomotion mechanism using four 2-way linear cylinders and four 2-way gripper modules. The robot may traverse a fixed distance or turn at a fixed angle after completing a series of repetitive cylinder and gripper actions termed gaits. Symmetric arrangement of the actuators decouples the translation and rotation of the robot. Hence the robot can swiftly change its direction of travel. Basic locomotion gaits are classified and modeled using finite state machines. Experiments on the real prototype are conducted to establish the repeatability and consistency of gait performances. Mechatronic design of the Planar Walker system is also described.
1
Introduction
In the past decade, there has been a strong interest in developing climbing and walking service robotic systems for construction, shipbuilding, chemical, petroleum, and nuclear industries. These robotic systems are able to move along structures or hulls of the facilities to perform maintenance or inspection tasks. Various locomotion methods have been devised for those systems (Backes et al., 1997, Hirose et al., 1997, Wang et al., 1999, Savall et al., 1999, Abderrahim et al., 1999, Hirose and Arikawa, 2000, Bevly et al., 2000, Unsal and Khosla, 2000, Longo and Muscato, 2001). In this article, we
describe a newly developed surface walking/climbing system for large surface inspection and maintenance tasks on ship hulls and oil tanks. A basic requirement on the system is to be able to traverse along the surface in a swift way with minimal turning radius so as to cover a very large working area in a short amount of time. Furthermore, the system design must be simple and robust for field operation, especially for ships at sea. A prototype system based on pneumatic power, called Planar Walker is constructed (Fig. 1). The Planar Walker can traverse on a plane or a smooth surface with a large radius of curvature. This system features a simple closed-loop planar 8-bar locomotion mechanism formed by four linear cylinders and four revolute joints. (Fig.2) The cylinders are not only used as actuating elements but also as chassis of the system. Hence, compact and lightweight system design can be achieved. When the four cylinders are actuated independently, the shape of the mechanism may change to a square, a rectangle, or an irregular quadrilateral. Four gripper modules (or suction pads) are mounted below each of the revolute joint to hold the robot to the working surface. To simplify the system control, the cylinders and grippers are designed to allow only two-way motions, i.e., on/off states (gripper) and fully retracted /extended states (cylinder). In this way, the robot may traverse (forward, backward, and sideway) a fixed distance or turn at a fixed angle after completing a series of repetitive cylinder and gripper actions termed gaits.
Due to the symmetric arrangement of the actuators, the translation and rotation of the robot are decoupled. Thus, it allows the robot to swiftly change its direction of travel. The purpose of this article is to introduce the locomotion principle and gait generation of the Planar Walker. We classify the basic locomotion gaits into traversing and turning gaits and model them using finite state machines. The finite state machine model enables us to study the gait generation and transition systematically. As the locomotion is a closed-loop mechanism, the validity of the gaits depends on the mobility of the effective mechanisms during the state transition. Experiments of the basic gaits applied to the actual prototype are conducted to establish the repeatability and continuity of the basic gaits. The results can be used as a basis for the navigation and path planning of the Planar Walker.
Gripper grips onto surface (fixed pivot)
N 2
1
3
4
Actuator (Retracted)
3
4
Actuator (extended)
E
S
(a) (b) Figure 2. Kinematic diagram of locomotion mechanism.
Figure 1. Prototype of Planar walker.
2
1
W
E S
N 2
W
Gripper released from surface (moving pivot)
Modeling of Locomotion Gaits
The locomotion of a robotic system is in general caused by the geometric phase changes due to its own body motion (Kelly and Murray, 1995, Murray et al., 1999). The locomotion of the Planar Walker is based on the shape-changing mechanism driven by the four independent cylinders and properly activated grippers. The four cylinders are identical. We assume that the robot can move only on a flat plane at present. Without loss of generality, we use the center distance of the two connected pivot joints to denote the length of the cylinder and denote the fully retracted and extended lengths as lmin and lmax respectively. The four gripper units can be activated perpendicular to the plane independently. They provide the friction needed for locomotion. Because the gripper unit is attached under the pivot joint, when activated, the associated joint becomes a fixed pivot joint. Otherwise, the pivot joint can move freely. Similar to inchworm-like systems (Chen et al., 2001), a cyclic action of the cylinders and the grippers that can re-position and re-orient the mechanism is called a gait. As the cylinders and grippers have 2-way actions only, we can use a finite state machine (Chen et al., 2001) to describe the robot gaits. DEFINITION
1:
The
state
of
actuators
in
the
Planar
Walker
is
an
8-tuple
q = ( xE , x1 , xN , x2 , xW , x3 , xS , x4 ) , where xi ∈{0,1} . Variables xE , xN , xW , and xS represent on/off states of the grippers (Fig.2). When xi = 0 , Gripper i is released; xi = 1 , Gripper i is engaged. Variables x1 , x2 , x3 , and x4 represent states of the cylinders. When xi = 0 , Cylinder i is fully retracted; xi = 1 , Cylinder i is fully extended.
DEFINITION 2: The gait of the Planar Walker is a sequence of actuator states {q0 , q1 , " , q f } such that q0 = q f .
Based on Definition 2, a gait has two features. First, it is initial state dependent. A gait represents a cyclic sequence of actuator states that goes back to the original state. Two gaits with different initial states will produce different types of motion. Second, the transition rule between any two states must follow the physical constraints imposed on the mechanical system. Algebraically, the on/off state of the gripper or the cylinder at any instance can be switched to a new state. However, the actual system does not allow the actuator state to be altered arbitrarily. For example, if two neighboring grippers are engaged with the surface, the cylinder in-between the two cylinder cannot be activated. One of the two grippers must be released before activating the cylinder. Once an initial state and the transition rule are given, the actuator states in-between can be obtained as follows. DEFINITION 3: The gait generator of Planar Walker is a 5-tuple A = (Q, Σ, δ, q0 , F ), where Q is a finite set of actuator states, Σ is a non-empty set of event labels called an input alphabet, q 0 ∈ Q is the initial state, F ⊆ Q is the set of final states, and δ is the transition function mapping δ : Q × Σ → Q . That is, δ (q, a) is a state for each state q and input symbol a ∈ Σ . The 5-tuple A is a finite automaton (Hopcroft and Ullman, 1979). Alternatively, A can be
represented by a directed graph in which the nodes are the states in Q, the arcs are the transition defined by the function δ, and the set of labels for the arcs are the alphabets in Σ. The gait generation problem becomes finding a path (a sequence of arcs) from the root node (denoted the initial state q0 ) to the desired node (denoted the final state qf ) as illustrated in Fig.3. Through this transformation, standard graph search algorithms can be utilized on the gait generation (Corman et al., 1990). As the length of the actuator state is 8, the total number of actuator states (or the size of Q) would be 28 = 256 - a fixed constant. Therefore, the complexity of the gait generation problem should be O(256( k − 2) ) with a given initial state, where k is the number of actuator states in a gait sequence. q0
δa q1
q2
......
qk
δb δc qf
qm
qn
qf2
Figure 3. Search on the graph formed by A.
3
Basic Gaits
If we indiscriminately select an initial actuator state and ignore the fact that the state transition must obeys physical constraints, the gait generation problem will end up with above-mentioned complexity. In reality, this is not so because of the symmetric structure in the system. There are only handful types of useful gaits that create symmetric body movements for locomotion purpose,
termed basic gaits. These gaits can be obtained in a heuristic manner. We will first introduce how these basic gaits work and then look at the generic state transition rules for the actuators. Due to symmetric design of the Planar Walker, there are only two types of gaits: the transverse gait and the turning gait. A transverse gait would enable the mechanism to travel forward (backward, left, or right) without changing the orientation. A turning gait would rotate the machine with a finite angle about its center of geometry in the clockwise or counter clockwise direction. 3.1 Transverse Gaits
The complete sequence of a transverse gait moving forward is shown in Fig. 4. We denote it as QTF 0 ≡ {qa , " , q j } . The locomotion mechanism does not have a head or tail portion due to its symmetric layout. The gait sequence of forward transverse motion is based on a northward direction of travel. Similar gait sequence can be generated for backward and sideway transverse gaits. As indicated in the figure, a transverse gait needs 10 steps to complete. Initially the cylinders are all retracted to form a square with side length of lmin and the grippers are all engaged with the surface as shown in Step a. To move toward north, Gripper N is released (Step b) and Cylinder 1 and 2 are to extend simultaneously (Step c). The shape of the robot changes from a square to a quadrilateral extending forward here. Then Gripper N is engaged again; Gripper E and W are released (Step d, e). After Gripper E and W are released, all cylinders are actuated to move Gripper E and W (or the body of the robot) forward (Step f). The shape of the robot is inverted from Step e to f. From Step h to Step i, Cylinder 3 and 4 are retracted to return to the shape of the original square. Finally Gripper S is engaged with the surface again (Step j). The state of the actuators goes back to the initial one and the robot moves forward with a fixed distance. Note that the orientation of the robot remains unchanged for this transverse gait. The actuator states of this transverse gait are given in Table 1.
2
1
3
4
b
S
3
4 S
N 2
W
2
W
3
h
3
4
1 E
E
E
e
S
3
4 S
3
f
4 S
N 2
1
1
W
E l
E 3
4 S
d
W
E
4
E
2 W
1
W
N 2
1
E
S
4 S
N 1
3
3
c
2
1
W
N
N
N 2
1
W
E
E
g
2
1
W
W
a
N
N
N 2
3
4
4 S
S
j
i
Figure 4. Transverse gait (forward, initially retracted). DEFINITION 4: The stride of a transverse gait (initially retracted) is the displacement of the center of the robot after completing one gait sequence. The stride length is
l=
lmax − 12 lmin − 2
2
1 2
lmin
(1)
The stride lengths of a transverse gait (initially retracted) moving backward or sideways are identical.
Remark 1: A transverse gait can also be initiated with all cylinders fully extended, i.e., q0 = (1, 1, 1, 1, 1, 1, 1, 1). The initial shape of the robot is a square with side length of lmax . The actuator sequence is shown in Fig. 5. The stride length is l=
1 2
lmax
2 2 − lmin − 12 lmax
Table 1. States in transverse gait (initially retracted). State
Transverse Gait (fwd)
QTF0 a b c d e f g h i j
(xE , x1, xN , x2 , xW , x3 , xS , x4 )
2
1
State
Turning Gait (CW)
QRR0 a b c d e f g h i j k l m
(xE , x1, xN , x2 , xW , x3 , xS , x4 )
M=2 M*=1 M=4 M*=2 M=2 M*=1
N
N 2
N 2
1
W
W
Table 2. States in turning gait (initially retracted).
Effective Mobility
(1, 0, 1, 0, 1, 0, 1, 0) (1, 0, 0, 0, 1, 0, 1, 0) (1, 1, 0, 1, 1, 0, 1, 0) (1, 1, 1, 1, 1, 0, 1, 0) (0, 1, 1, 1, 0, 0, 1, 0) (0, 0, 1, 0, 0, 1, 1, 1) (1, 0, 1, 0, 1, 1, 1, 1) (1, 0, 1, 0, 1, 1, 0, 1) (1, 0, 1, 0, 1, 0, 0, 0) (1, 0, 1, 0, 1, 0, 1, 0)
(2)
(1, 0, 1, 0, 1, 0, 1, 0) (1, 0, 0, 0, 1, 0, 0, 0) (1, 0, 0, 1, 1, 0, 0, 1) (1, 0, 1, 1, 1, 0, 1, 1) (0, 0, 1, 1, 0, 0, 1, 1) (0,1, 1, 1, 0, 1, 1, 1) (1,1, 1, 1, 1, 1, 1, 1) (1, 1, 0, 1, 1, 1, 0, 1) (1, 0, 0, 1, 1, 0, 0, 1) (1, 0, 1, 1, 1, 0, 1, 1) (0, 0, 1, 1, 0, 0, 1, 1) (0, 0, 1, 0, 0, 0, 1, 0) (1, 0, 1, 0, 1, 0, 1, 0)
N 2
1
M=4
M=4
M=4
M=4
N 2
1
W
W
Effective Mobility
N
2
1
1
W W
E
E 3
3
4 S
a
4
E
4
S
S
b
3
3
c
S
d N
N
2
1
E
W 3
g
1
S
2
W
e
3
i
4 S
E 3
E
f
4 S
1
W
E
4
3
N
E 3
h
2
1
W
4 S
N
2
E
4
E 3
4 S
j
4 S
Figure 5..Transverse gait (forward, initially extended).
3.2 Turning Gaits
Figure 6 shows the execution sequences of a clockwise turning gait. We denote it as QRR 0 ≡ {qa , " , qm } . A counterclockwise turning gait can be deduced with similar analogy. The turning gait needs 13 steps to complete one cycle. The cylinders are all retracted to form a square with side length of lmin initially and the grippers are all engaged with the surface (Step
a). Gripper N and S are released to initiate the turn (Step b). Cylinder 2 and 4 are extended simultaneously (Step c). The shape of the robot changes from a square to a parallelogram here. Then Gripper N and S are engaged with the surface; Gripper E and W are released (Step e). With diagonal grippers N and S engaged, Cylinder 1 and 3 are fully extended at the same time (Step f). The shape of the robot becomes a square with side length of lmax . Compared with the initial orientation of the robot, the robot has turned a fixed angle. However, the states of the actuators are all fully extended, and have not returned to the original states yet. The second stage of rotation starts from Step g and finishes at Step m. The states of the diagonal grippers or opposite cylinders change alternatively to complete the turning motion. The shape of the robot changes from the large square to a parallelogram and to the original small square at last. The robot has rotated clockwise with angle α about its center relative to the original configuration. Note that the center of the robot remains unchanged after the turns. The actuator states of the turning gait are given in Table 2. 2
1
1
3
a 2
4
c
E
E
4
3
d
W
W
E
4 S
e
S
1
3
4
3
N
2
1
2
W
3
S
b
N 1
2
W
E
E 4 S
N 1
2
W
3
W
N
N
N 2 W
2
W
N
2
W
N
1
1
W
2
N
1
3
3 S
3
E
f
4
g
S
W
1
2
N
W
2
E
S
N
1
h
1
4
S
N
α
N
4
2
W
E
3 S
i
4
E
3
3 S
j
4
E
1
4
S
S
E
k
3
4
l
E
S
4
E
m
Figure 6. Turning gait (clockwise, initially retracted). DEFINITION 5: The rotation of a turning gait (init. retracted) is the angular displacement of the robot’s orientation after completing one gait sequence. The angle of rotation is
lmax − lmin 2
α = 2 sin
−1
2
lmax lmin
(3)
The angle of rotation of a counter clockwise turning gait is also α.
Remark 2: The turning gait re-orients the robot about its center of geometry and is decoupled from the displacement of the transverse gaits. Therefore, the robot can do zero-radius turn at any position. When navigating the Planar Walker according to a path or a goal point, the orientation and position of the robot can be planned individually. Remark 3: A turning gait can also be initiated when all cylinders are fully extended, i.e., q0 = (1, 1, 1, 1, 1, 1, 1, 1). In this case, the initial shape of the robot is the square with side length of lmax . It can be shown that the angle of rotation of such gait is also α. 3.3 Summary of Basic Gaits
We use Table 3 to summarize the basic gaits of the Planar Walker. We know the gaits are initialstate dependent. Therefore, the basic gaits can also be categorized into two groups: one with fully retracted initial states and one with fully extended initial states.
Gaits with identical initial states can be used alternatively right away because the final state of the prior gait is identical to the initial state of the subsequent gait. A series of transitional states is necessary for using gaits with different initial states. This series of transitional states can be obtained using the gait generator mentioned in Definition 4. For example, if the Planar Walker want to change the type of gaits from the fully retracted state to the fully extended state, it may go through the actuator states from a to g of Fig. 6 (Turning gait). Afterwards, the robot can use any basic gaits in the category of QTF1, QTB1, QTR1, QTL1, QRR1, or QRL1. Table 3. Notations and pose changes of gaits.
QTF0 QTB0 QTR0 QTL0 QRR0 QRL0 4
Transverse Gaits fwd, retracted QTF1 fwd, extended bck, retracted QTB1 bck, extended right, retracted QTR1 right, extended left, retracted QTL1 left, extended Turning Gaits cw, retracted QRR1 cw, extended ccw, retracted QRL1 ccw, extended
TY TY-1 TX TX-1 RZ-1 RZ
Validation of Gaits
As mentioned in Section 2, the transition of the actuator states in a gait must follow the physical constraints of the system. Otherwise, the gait becomes invalid and cannot be implemented physically. The transition between two neighboring states, i.e., qi → qi +1 , are related to either the gripper or the cylinder state changes. 4.1 Gripper State ( xE , xN , xW , xS ) Change
Grippers represent the pivoting position of the robot on the surface. For any state transition, at least TWO identical grippers must be engaged with the surface. Otherwise, the robot may not hold on to its current position. 4.2 Cylinder State ( x1 , x2 , x3 , x4 ) Change
Cylinders represent the actuators of the mechanism. Hence, any change in the cylinder states must obey mobility of the mechanism, i.e., the cylinder actions must be kinematically viable. Although the 8-bar locomotion mechanism is designed with four linear actuators, not all of them will be actuated simultaneously during the complete gait cycle. In addition, the fixed pivot points changes according to the gripper actions. Therefore, the effective number of rigid links and joints of the locomotion mechanism during any state transition would be changing all the time. Basically, the mobility of the effective locomotion mechanism during the state transition must follow the Kutzbach criteria. For a planar mechanism, the Kutzbach criterion reads as M = 3 ( n − 1) − 2 J 1 − J 2 , (4)
where M is the mobility of the effective locomotion mechanism; n is the effective number of links; J1 and J2 are the effective number of 1-DOF and 2-DOF joints respectively. We can use the following theorem to verify the actuator state transition based on cylinder state changes. THEOREM 1: Let Mi be the mobility of the effective locomotion mechanism and Ni the effective cylinder actuators during the gait transition qi → qi +1 respectively. The gait of the Planar Walker is valid if and only if M i ≤ N i for all i.
Remark 4: When Mi < Ni, the effective locomotion mechanism is said to have redundant actuators. In this situation, the robot is still operable with coordinated actuator movements and is also fault-tolerant to actuator failures. Transverse gaits. As shown in Fig.4, from Step b to c (or Step h to i), the mechanism is essentially equivalent to a 5-bar linkage (with Gripper E, W S engaged). The mobility of the effective mechanism is 2, matching the number of effective cylinders (1 & 2). From Step e to f, the mechanism is equivalent to a 9-bar linkage (with Gripper N, S engaged). The mobility is then 4, which matched the number effective cylinders (1, 2, 3, & 4). Turning gaits. As shown in Fig.6, from Step b to c (Step e to f; Step h to i; Step k to m), the mechanism is equivalent to a 9-bar linkage (with diagonal grippers engaged). The mobility of the mechanism is 4, which matches the number of the effective cylinders (1, 2, 3 & 4). The effective mobility of the locomotion mechanism during the gait transition for the forward transverse gait and the clockwise turning gait are indicated as M in Table 1 and 2. Invalid transverse gait. To illustrate how the mobility affects the movement of the locomotion mechanism, consider the transverse gait depicted in Fig.7. Intuitively this gait is achievable. However, from Step b to c (or Step e to f), the mechanism is equivalent to a 7-bar linkage with mobility of 4, which is more than the number of the effective actuating cylinders 3 (Cylinder 1,2, & 4). The mechanism thus has undetermined DOF and may not end up with the configuration of Step c (or Step f). Hence, this is an invalid gait. N N 2
N 2
1
W
W
1
2
W
W
a
E 3
4 S
b
S
c
d
S
4
3 S
e
E 3
4
3 S
f
1
W
E
E
4
3
4
2
1
W
W
E E 3
2
1
2
N
N
N
N
1
2
1
4 S
l* g
E 3
4 S
l*
Figure 7. Invalid transverse gait.
5
Mechatronic System Design
A prototype of the surface walking robot system is designed and constructed for feasibility study (Wong, 1999). The prototype is compact in size (within 50cm x 50cm) and the weight is less than 6 kg. The robot system consists of two major modules: Locomotion mechanism module and System control module. As we are investigating the feasibility of the walking mechanism, the service module based on the mission requirement has not been deployed yet. 5.1 Locomotion Mechanism
Figure 8 shows the mechanical arrangement of the locomotion mechanism from below. Four pneumatic cylinders are connected through carefully designed pin joints. This joint design allows
unrestricted joint motion for all gait sequences. The cylinders have a bore size of 16mm with a stroke length of 45mm. The overall length of the pistons with two pin joints is 175.5mm. The piston configuration allows a turning angle of 25° and a transverse stride of 32mm.
Gripper Cylinder
Figure 8. Underside view of Planar walker.
Figure 9. Locomotion mechanism with crossbar frame.
To provide greater support to the 4-cylinder structure, a crossbar frame is added. This crossbar frame serves as a structural frame to strengthen the locomotion mechanism, to facilitate the mounting of idle load-supporting wheels, and to act as the mounting platform for the controller, peripheral devices and the service module. The crossbar frame has a center revolve joint so that the two crossbars can rotate with respect to each other with no restriction. The crossbar frame consists of four sliding units that are integrated with the pin joints between the cylinders. To prevent undesirable rotation of the bearing support of the sliding unit, a special toothed shape slider shaft design is selected. A total of five idle wheels are placed under the crossbar frame to provide stability and support the robot moving on a plane. The kinematic diagram of the locomotion mechanism with the addition of the crossbar frame is shown in Fig.9. Because of this modification, the mobility of the locomotion mechanism and its effective mechanism during gait transition will be different from the original ones. This modified mobility M* for the effective mechanism during gait transitions are indicated in Table 1 & 2. Note that the reduced mobility actually provides fault-tolerance for actuator failure. A gripper unit is attached to each of the slider unit to provide the vacuum suction force to the surface. The gripper unit has a vacuum pad and a lift actuator. Suction between the robot and the surface is attained when the vacuum pad encloses the ground with the application of vacuum. The vacuum pad is lifted when the robot is moving. 5.2 System Control Module
The system control module consists of a PC data acquisition card, signal amplification circuit, an LED debug/monitoring display unit, and a motion/gait control program. The data acquisition card has 16 5V-output channels. A total of 12 channels are used to control the 12 solenoid valves: 4 for the motion actuation cylinders; 4 for the pad lifting actuators; 4 for the on/off control of the air supply to the four vacuum generators. A C-language program Pathwise is developed for motion control and gait generation (Ting, 2000).
6
Experiment Observation and Result
6.1 Preliminary System Test
The objective of this test is to see whether the robot can produce the desired basic gaits according to the given commands and its locomotion capability on flat ground and the slope. The result of the test is also served as the basis for further fine-tuning of the actuators and motion control elements. Figure 10 shows a series of snap shots of the Planar Walker in motion. The following observations are noted from the preliminary tests carried out on the Planar Walker:
(a)
(b)
(c)
(d)
Figure 10. Planar Walker in motion.
– The prototype takes 30 sec to execute a transverse gait and 55 sec for the turning gait. – When placed on an inclined plane made of perspex, the robot is able to travel successfully in all directions and execute turning movements at an incline angle of up to 30o. At greater incline angle, it starts slipping during certain gait. The ability to travel on inclined surface demonstrates the viability of a wall-climbing model based on the present prototype. – A slightly jerky motion is encountered initially when the rates of extension and retraction between the four cylinders are different though the end positioning is not affected. However, with proper tuning and adjustment, the transitional motion was brought to a smoother level. 6.2 Gait Performance Evaluation
To verify whether the actual stride lengths and angle of turns meet the design specifications mentioned in Section 5.1 or not, we conducted experiments on the individual gaits to establish the actual performance. Speed of the gaits. The speed of executing a complete transverse gait or a turning gait depends mainly on the speed of the cylinder movements. The speed of the cylinder is basically a function of the air pressure supplied to the system, flow rate of the air supplied to the cylinder, and the switch time of the solenoid valves. Supplying large air pressure (8.5 bar max) to the robot could increase the speed of the gaits but also destabilize the gait pattern because the extension and contraction speeds of the four cylinders are not uniform. To ensure smooth movement of the cylinders and hence the gaits, fine-tuning of the cylinder speed is done manually. After finetuning, the speed of a transverse gait is brought to 30 sec/cycle and a turning gait 60 sec/cycle.
Repeatability of gaits. The stride length of the transverse gait is established in two manners: repeatability and continuity. For repeatability, we repeat the operation of a single gait 8 to 10 times and measure the displacement and orientation change of the gait each time. All basic gaits with fully retracted initial states (Transverse – forward, backward, right, and left; Turning – CW and CCW) are measured. Table 4 and 5 show the result of the repeatability of the gaits. Table 4. Repeatability of transverse gaits. Forward Backward Right Left S 1 2 3 4 5 6 7 8 9 10
34 33.5 33.5 33 33 33 12 33 34 33
Avg 31.2
θ
S
θ
S
θ
