Analysis and solution for the curling stone throwing of a novel six-legged curling robot in curling competition

Abstract

In curling competitions, the throwing strategy has a decisive influence on the outcome of the game. When robots are applied to the sport of curling, they first need to understand the various throwing strategies in curling competitions and then adjust their motion control parameters to achieve the corresponding strategic throws. However, current curling strategy research lacks mathematical analysis and descriptive methods for throwing strategies tailored to robots. Moreover, research on how robots can solve for corresponding throwing strategies is lacking. These limitations have restricted the application and development of curling robots in the sport. Here, the concepts of the curling stone’s hitting domain and hitting tree are introduced to analyze and describe the curling strategies for robots by constructing the curling hitting domain through a curling collision model and by building the hitting tree through operations such as combination, permutation, and pruning. Furthermore, based on the solution methods for hitting domains and hitting trees, a search solution method for the control parameters of robots is developed. The research findings are integrated into a curling robot auxiliary decision-making software. With the help of the auxiliary software, the curling robot achieves victory in competitions against humans. The research outcomes are of great importance for the application and development of curling robots and legged robots.

Graphical abstract

Keywords

curling robot / curling strategy / legged robot / control parameters

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Nomenclature

Abbreviations
AI	Artificial intelligence
KR-UCT	Kernel regression-upper confidence bound applied to trees
MCTS	Monte-Carlo tree search
PTFE	Polytetrafluoroethylene
RL	Reinforcement learning
Variables
ai	Coefficients of a seven-degree polynomial trajectory curve (i = 0,1,2,...,7)
acs	Acceleration of the robot and the stone
ar	Curling stone’s radial acceleration
at	Curling stone’s tangential acceleration
d	Gliding distance of the robot in this stage
dk	Movement distance of the robot kicking
dp	Distance of the robot pushing stone
Ev	Overall kinetic energy of the robot and stone
Ef	Work done by the friction at this stage
F	Friction force
Fb	Frictions of the back parts of the curling stone
Fbr	Radial frictions of the back parts of the curling stone
Fbt	Tangential frictions of the back parts of the curling stone
Ff	Frictions of the front parts of the curling stone
Ffr	Radial frictions of the front parts of the curling stone
Fft	Tangential frictions of the front parts of the curling stone
Fr	Radial frictions of the curling stone
Ft	Tangential frictions of the curling stone
I	Inertia moment of the curling stone
m	Curling stone mass
mA	Mass of curling stone A
mb	Robot mass
mB	Mass of curling stone B
mc	Mass of curling stone
mf	Mass of the front parts of the curling stone
n	Total number of curling stones on the ice
Ob1	Center of the robot body before angle adjustment
Ob1−xyz	Robot body coordinate frame
Ob2	Center of the robot body after angle adjustment
OL−xyz	LiDAR coordinate system
OM	Center of the tips of the rear legs
OW−xyz	World coordinate system
P	Position vector of the front/back part of the curling stone
Pb1	Position of Ob1 in the world coordinate system
b1Pb2	Robot body position in the robot body coordinate frame (Ob1−xyz)
PNM	Position vector from curling stone N to curling stone M
Ptipi	Positions of the tips of each leg in the world coordinate system
qr	Real-time position of the robot
q(t0)	Robot position at the beginning of the kicking stage
q(t1)	Robot position at the end of the kicking stage
r	Rotation radius
r	Position vector of the front/back part of the curling stone with respect to the center of the curling stone
R	Position vector of the center of the curling stone
WRb1	Robot body posture in the world coordinate frame before the angle adjustment
SDD″	Distance from point D to point D″
t	Real time
t0	Time of the beginning of the kicking stage
t1	Time of the end of the kicking stage
t2	Time of the end of the gliding stage
t3	Time of the end of the pushing curling stone stage
tg	Glide time of the robot and the stone
vb	Curling robot velocity
vc	Curling stone velocity
vcs	Velocity of the robot and the stone
vk	Robot’s velocity after kicking
vp	Forearm push velocity reach
vr	Radial velocity component
vt	Tangential velocity component
vX	Speed of stone at the time of the collision (X = A,B)
vXn	Normal velocity component of stone at the time of the collision (X = A,B)
v_{X\text{n}}^{\prime }	Normal velocity component of stone after collision (X = A,B)
vXt	Tangential velocity component of stone at the time of the collision (X = A,B)
v_{X\text{t}}^{\prime }	Tangential velocity component of stone after collision (X = A,B)
VD	Stone D collision velocity
VD′′	Velocity of the stone D at point D′′
VX′	Speed of stone after collision (X = A,B)
θ	Target angle
φ	Angle between the curling stone’s central position vector and the polar axis
ω	Rotational angular velocity
μ	Friction coefficient between curling stone and ice surface
μAct	Actual friction coefficient between curling stone and ice surface
μb	Friction coefficient between the curling stone back part and ice surface
μc	Friction coefficient between the stone and the ice
μCal	Calculated friction coefficient between curling stone and ice surface
μf	Friction coefficient between the curling stone front part and the ice surface
\lambda	Angle between tangential velocity and radial velocity
{\Phi }_{A{B}_1}	Angle of the position vector P_{A{B}_1}
{\Phi }_{A{B}_2}	Angle of the position vector P_{A{B}_2}
{\Phi }_{B_{1}B}	Angle of the position vector P_{B_{1}B}
{\Phi }_{B_{2}B}	Angle of the position vector P_{B_{2}B}
{\Phi }_{NM}	Angle of the position vector P_{NM}

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 92248303).

Electronic Supplementary Material

The supplementary material is available in the online version of this article at https://doi.org/10.1007/s11465-025-0835-5.

Conflict of Interest

The authors declare no conflict of interest.

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