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Page 112 Yang et al. Intell Robot 2024;4(1):107-24 I http://dx.doi.org/10.20517/ir.2024.07

horizontal and vertical supporting forces from the support leg and are expressed as follows:

{

= ¥ com ( + + ) + , (2)

= (¥ com + )( + ) − ,
where , and represent the masses of the human subject, the robotic walker, and the exoskeleton
robot, respectively. ¥ com and ¥ com are the second derivatives of the COM’s position com and com. is the

constant gravitational acceleration. The torques of the hip and knee joints of the exoskeleton support leg can
be calculated as follows:
[ ] [ ] [ ] [ ]
cos( ) sin( )
= , (3)
0 cos( − ) sin( − )
where and are the torques of the hip and knee joints. As a result, the power of the exoskeleton robot is
determined by the joint torques and and the angular velocity of the joint and . Let us take the hip
¤
¤
joint as an example; the power of the hip joint’s motor is calculated as:
2
 = · + · ,
¤


= / = · , (4)

= / ,
 ¤ ¤

where and represent the torque and angular velocity of the motor in the hip joint, respectively. repre-
¤
sents the current of the motor, and is the reduction ratio. Besides, and represent the resistance and
torque constant of the motor, respectively. The first item of in Equation (4) represents the thermal power,
while the second item indicates the mechanical power. With the power of the motors in the joints of the sup-
port leg, the energy consumption of the exoskeleton support leg during the stance phase can be calculated as
follows:
∫
= ( + ) , (5)
0
where and denote motor power of the hip and knee joints, respectively. signifies the duration of the

stance phase in one gait cycle.
Based on the Equations (2)-(4), we can find that the torques of the hip and knee joints are decreased as the
supporting force increases, i.e., the energy cost of the exoskeleton robot is decreased with increasing .
However, if the supporting force is too large, the human-exoskeleton system will be lifted off the ground, and
the friction between the ground and the exoskeleton’s foot will be reduced, resulting in an abnormal walking
posture of the human-exoskeleton system and even with slipping over the ground. Therefore, finding the
appropriate supporting force to minimize the energy consumption and prevent slipping is critical. Now, let us
construct an objective function to denote the energy efficiency:

(·) = / , (6)

where denotestheenergyconsumptionofbothhipandkneejointsofthesupportlegduringthestancephase
in one gait cycle; represents the stepping length for one step. Consider the value of the objective function
as Total Cost of Transport (TCoT). Now, let us find a way to solve the objective function and find the optimal
supporting force.

2.1.2 Real-time optimization of the supporting force
In this subsection, the real-time optimization of the supporting force, which employs the discrete-time ESC
approach, is presented. ESC is a model-free adaptive control method that finds an optimum set-point in order
to minimize/maximize an objective function, whose analytical expression might be unknown [20–23] . Kumar
et al. proposed a modified structure of the discrete-time ESC by introducing a stepper motor with an integra-
tor [24,25] . In this modified structure, the ESC integration is performed by the motor dynamics itself. Moreover,