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Page 4 of 16 Wang et al. Intell Robot 2023;3(3):479-94 I http://dx.doi.org/10.20517/ir.2023.26

Assumption 2.3. The swinging leg did not slip or rebound with the ground during the collision.

Remark 1: The Assumptions 2.1-2.3 are general. These assumptions make it necessary to establish a relation-
ship between the walking process and the dynamic model. Additionally, these assumptions have also been
used in [29,30] .

The motion equation in the double support phase is described as,

( ) ¥ + ( , ¤) ¤ + ( ) = ( + ) (1)

where ( ) ∈ × is the positive-definite inertia matrix, ( ) = / ∈ is the gravity matrix, and
= Φ/ ∈ is the Jacobian matrix. Φ represents robot constraints, and Φ( ) = 0. The nonlinear

dynamic equation of the biped robot can be described as a second-order differential equation. ( , ¤) ∈ ×
is the centrifugal force and Coriolis force terms, and

( 1

( , ¤) ¤ = ( ( ) ¤) ¤ − ( ( ) ¤) ¤
0
2
Í 1 (2)
= + − , ≥ 1, ≤
=1 2
where N is the length of the generalized configuration vector. ∈ is the input torques, and ∈ is

the external disturbance. Considering the external force exerted on the robot foot during an impact, which is

¯ +
defined as ext = ext ( ) ( ), (1) can be described as
−

( ) ¥ + ( , ¤ ) ¤ + ( ) = +
(3)
−
¤ + ¤ − ¤ − ¤ =
+

where e = 1 2 , and , shows the hip position in Cartesian coordinates. ( e ),
( e ), and ( e ) are inertia matrix, Coriolis force matrix, and gravity matrix in the double support phase,
respectively. The collision mapping can be written as

= − (4)
+
+ −
¤ = Δ ( ¤ )
Describing (2) and (4) as the form of state space, as shown in (5), demonstrates that the walking system is
hybrid.

¤ ( ) = ( ( ( )) + Δ ( ( ))) + ( ( ( )) + Δ ( ( ))) ( ) ( ) ∈ \
Σ : (5)
+
−
−
( ) = Δ ( ( )) ( ) ∈
−
+

where ( ) = [ , ¤] is the defined state variable, and ( ) = [ , ¤ ] and ( ) = [ , ¤ ] represent state

+
+
−
−
variables before and after the impact, respectively. ( ( )) and ( ( )) are bounded nominal nonlinear func-
tions; Δ ( ( )) and Δ ( ( )) represent uncertainties. Through differential homeomorphism transformation,
the continuous part of the hybrid system (5) can be expressed as a nonlinear system with uncertainties,