IEEE Transactions on Control Systems Technology · 2026

Turning Circle-based
Control Barrier Function
for Efficient Collision Avoidance
of Nonholonomic Vehicles

Changyu Lee¹, Kiyong Park², and Jinwhan Kim^2,*

¹Kongju National University · ²KAIST

Changyu Lee PRESENTER

Assistant Professor, Department of Mechanical & Automotive Engineering, Kongju National University

leeck@kongju.ac.kr

Background · MPC-based obstacle avoidance

Distance constraints in MPC (MPC-DC)

Vehicle with safety radius R_s at position (x, y) and a circular obstacle of radius o_r; h(x) is the clearance between them

Distance constraint (obstacle avoidance): \(h(\mathbf{x})\ge 0\)

MPC-DC

\(i=0,\ldots,N-1\)

\(\displaystyle\min_{\mathbf{x}(\cdot),\mathbf{u}(\cdot)}\) \(\displaystyle\sum_{i=0}^{N-1}\) \(\Big(\) \(\|\mathbf{x}_i-\mathbf{r}_i\|_Q^2\)Tracking errorreference path \(+\) \(\|\mathbf{u}_i\|_R^2\)Control effort \(\Big)\) \(+\) \(\|\mathbf{x}_N-\mathbf{r}_N\|_P^2\)Terminal cost

\(\text{s.t.}\) \(\mathbf{x}_0-\mathbf{x}_{\mathrm{init}}=0\) Initial condition \(\mathbf{x}_{i+1}-f_d(\mathbf{x}_i,\mathbf{u}_i)=0\) Vehicle dynamics \(\mathbf{x}_i\in\mathbf{X},\ \ \mathbf{u}_i\in\mathbf{U}\) State and input limits \(h(\mathbf{x}_i)\ge 0\) Distance constraintsafe clearance at every step

Safe distance constraints can be incorporated into an MPC's state constraints to generate collision-free trajectories

Background · Limitation of MPC-DC

Distance constraints need a long horizon

With a long prediction horizon the predicted path bends around the obstacle; with a short horizon the prediction ends before the obstacle

Horizon 1.5 sstuck at the obstacle

Horizon 3.0 sreaches the goal

Same scenario — only the prediction horizon of MPC-DC differs

Obstacle avoidance constraints are only activated when the vehicle is close to obstacles
→ MPC-DC requires a long prediction horizon, which increases the computational load

Background · CBF-based MPC [1]

CBF constraints in MPC (MPC-CBF)

Exponential CBF: the rate of change of the clearance h toward the obstacle, h-dot, is bounded by the clearance itself

Under the CBF condition h-dot is at least minus alpha times h, so h may decrease but always stays positive

MPC-CBF same cost and constraints as MPC-DC, except the obstacle constraint

\(\text{s.t.}\) \(\mathbf{x}_0-\mathbf{x}_{\mathrm{init}}=0,\ \ \mathbf{x}_{i+1}-f_d(\mathbf{x}_i,\mathbf{u}_i)=0,\ \ \mathbf{x}_i\in\mathbf{X},\ \mathbf{u}_i\in\mathbf{U}\) \(\Delta h_e(\mathbf{x}_i,\mathbf{u}_i)+\alpha_e h_e(\mathbf{x}_i)\ge 0\) Discrete-time CBFreplaces \(h(\mathbf{x}_i)\ge 0\) \(\Delta h_e(\mathbf{x}_i,\mathbf{u}_i)=h_e(\mathbf{x}_{i+1})-h_e(\mathbf{x}_i)\) One-step change \(h_e(\mathbf{x})=\dot h(\mathbf{x})+\alpha\,h(\mathbf{x})\ge 0,\ \ \alpha>0\) Exponential CBFof the distance \(h\)

The CBF limits the speed toward an obstacle in proportion to the relative distance from it
MPC-CBF with a shorter horizon can still achieve performance similar to MPC-DC [1]

Motivation · Euclidean distance-based CBF (ED-CBF)

ED-CBF overlooks heading and turning

ED-CBF value map: a vehicle moving in +x at 1.5 m/s; a wide crescent in front of the obstacle has negative CBF values

The upper and lower parts of the crescent are highlighted green: states from which the vehicle could still turn away

1

Only the closest Euclidean distance

Conventional CBFs measure the shortest distance to the obstacle — the ED-CBF

2

Heading and turning are ignored

Limited turning capability is a common characteristic of nonholonomic vehicles — cars, robots, ships

3

Safe states are perceived as risks

States that are actually safe for navigation are restricted → conservative trajectories

Proposed

Nonholonomic characteristics in the CBF design → Turning Circle-based CBF (TC-CBF)

Proposed method

Turning Circle-based CBF (TC-CBF)

Clearance is measured from the vehicle's left and right turning circles

1

Two turning circles from the speed \(u\) and max. turning rate \(r_{\max}\)

\(R=u\,/\,r_{\max},\qquad \mathbf{p}_{tl},\,\mathbf{p}_{tr}=\mathbf{p}\pm R\,\big[\cos(\psi+\tfrac{\pi}{2}),\,\sin(\psi+\tfrac{\pi}{2})\big]^{\top}\)

2

A CBF for each circle: circle-to-obstacle clearance

\[\begin{aligned}h_{tl}(\mathbf{x})&=\big\|\mathbf{p}_{tl}-\mathbf{o}\big\|-(o_r+R+R_s)\\ h_{tr}(\mathbf{x})&=\big\|\mathbf{p}_{tr}-\mathbf{o}\big\|-(o_r+R+R_s)\end{aligned}\]

3

Safe while at least one turning circle does not intersect the obstacle

\(h_t(\mathbf{x})=\max\big(h_{tr}(\mathbf{x}),\,h_{tl}(\mathbf{x})\big)\)

implemented with a smooth log-sum-exp approximation of the max

Always leaves space on the left or right to avoid obstacles
In MPC as a discrete-time CBF: \(\Delta h_t(\mathbf{x}_i,\mathbf{u}_i)+\alpha_t\,h_t(\mathbf{x}_i)\ge 0\)

Proposed method · Comparison of restricted states

TC-CBF restricts far fewer states

ED-CBF value map: a wide crescent in front of the obstacle is restricted

TC-CBF value map: only a narrow lens-shaped region directly in front of the obstacle is restricted

Less conservative! Vehicle heading +x at 1.5 m/s · only states from which neither turn avoids the obstacle are restricted

Simulation · Unicycle · Static obstacle

ED-CBF stays conservative after tuning

Even with different parameter sets, ED-CBF generates more conservative trajectories than TC-CBF due to its inherent characteristics

MPC-TCCBF MPC-EDCBF \((\alpha,\ \alpha_e)\): (0.25, 0.05) (0.50, 0.05) (0.75, 0.05) (1.00, 0.05) (0.50, 0.03) (0.50, 0.07)

x-y trajectories around a static obstacle: MPC-TCCBF (red) deviates least; all six MPC-EDCBF variants detour further

Speed u: MPC-TCCBF keeps 2.0 m/s while every MPC-EDCBF variant slows to 1.2–1.4 m/s

TC-CBF: no slowdown

Turning rate r: MPC-EDCBF variants saturate at −0.2 rad/s early in the avoidance, MPC-TCCBF turns more gently

CBF values over time for all controllers stay non-negative

Simulation · Unicycle · Head-on encounter

Head-on: avoidance without slowing down

TC-CBF starts a gentle turn early instead of braking hard — the oncoming obstacle is avoided at full speed

Moving obstacle approaching head-onobstacle moves from x = 25 m toward the vehicle (0 → 30 s)

MPC-EDCBFMPC-TCCBF

Head-on scenario trajectories: both controllers pass the oncoming obstacle; snapshots every 2.5 s

Speed: MPC-EDCBF slows to about 1.2 m/s, MPC-TCCBF stays near 2.0 m/s

ED-CBF slows to ≈1.2 m/s

ED-CBF saturates at −0.2 rad/s

CBF values: the TC-CBF keeps a larger margin than the ED-CBF, both non-negative

TC-CBF keeps a margin

Simulation · Unicycle · Overtaking

Overtaking: a smaller detour, no slowdown

MPC-TCCBF passes with about half the lateral deviation (≈1.3 m vs ≈2.7 m) while keeping the desired speed

Overtaking a slower obstacleobstacle moves in +x along the path (0 → 20 s)

MPC-EDCBFMPC-TCCBF

Overtaking trajectories: MPC-TCCBF deviates about 1.3 m from the path, MPC-EDCBF about 2.7 m

Speed: MPC-EDCBF slows to about 1.4 m/s, MPC-TCCBF stays at 2.0 m/s

ED-CBF slows to ≈1.4 m/s

ED-CBF saturates at ±0.2 rad/s

CBF values of both controllers stay non-negative

Simulation · Summary

MPC-TCCBF is more efficient

Scenario	Controller	Arrival time \(t_a\) [s]	Speed error \(e_{\mathrm{speed}}\)	Cross-track error \(e_{\mathrm{cte}}\)
Static	MPC-EDCBF	21.6	0.088	1.273
Static	MPC-TCCBF	20.4	0.005	0.962
Head-on	MPC-EDCBF	26.9	0.107	0.889
Head-on	MPC-TCCBF	25.5	0.019	0.659
Overtaking	MPC-EDCBF	21.3	0.087	0.916
Overtaking	MPC-TCCBF	20.1	0.002	0.450

\(t_a\) arrival time · \(e_{\mathrm{speed}}\) average speed tracking error · \(e_{\mathrm{cte}}\) average cross-track error

5.5%↓

Arrival time

91%↓

Speed tracking error

34%↓

Cross-track error

vs MPC-EDCBF · average over the 3 scenarios

MPC-TCCBF is more efficient, avoiding unnecessary slowdowns and aggressive turns

Experiment · Setup

Real-world validation with an ASV

Clearpath Robotics Heron Clearpath Robotics Heron autonomous surface vehicle on the water

Experimental site · KAIST Aerial view of the pond at KAIST used for the experiments

VEHICLE

Differential-thrust catamaran ASV1.35 × 0.98 × 0.32 m · desired speed 0.9 m/s

ASV MODEL

\(M\dot{\nu}+C(\nu)\nu+D(\nu)\nu=\tau_c,\ \ \dot{\eta}=R(\psi)\nu\)\(\nu=[u,v,r]^\top\) velocities · \(\eta=[x,y,\psi]^\top\) pose

CONTROLLERS · SCENARIOS

MPC-TCCBF / MPC-EDCBF: \(N=20,\ T_s=0.1\) sMPC-DC: \(N=50,\ T_s=0.2\) s · static, head-on, overtaking

Experiment ① · Static obstacle

Smooth avoidance with a short horizon

Heron trajectories around a static obstacle for MPC-TCCBF, MPC-EDCBF and MPC-DC with time stamps every 6 s

Speed over time: MPC-DC swings between 0.6 and 1.2 m/s, MPC-EDCBF slows to about 0.65 m/s, MPC-TCCBF stays near 0.9 m/s and arrives first

Closest distance to the obstacle over time for the three controllers, all non-negative

MPC-DC N = 50, Ts = 0.2 s

Aggressive maneuvers when close to the obstacle, even with a long horizon

MPC-CBF N = 20, Ts = 0.1 s

Both CBF controllers avoid smoothly with a significantly shorter horizon

MPC-TCCBF vs MPC-EDCBF

Arrival time 10%↓ · speed error 68%↓ · cross-track error 8%↓

Experiment ② · Head-on encounter

Head-on: faster, closer to the path

MPC-TCCBF proposed

051015202530 05101520 x [m]y [m]

TC-CBF values over time: decreasing from 16 to a minimum near 1 and rising again, always positive

MPC-EDCBF

051015202530 05101520 x [m]y [m]

ED-CBF values over time: decreasing from about 4 to a minimum near 0.3 and rising again, always positive

~~42.2 →~~38.8s

Arrival time \(t_a\)8%↓

~~0.097 →~~0.051

Speed error \(e_{\mathrm{speed}}\)47%↓

~~2.014 →~~1.453

Cross-track error \(e_{\mathrm{cte}}\)28%↓

Both CBF values stay positive — collision-free in both runs

Experiment ③ · Overtaking a moving obstacle

Overtaking: experiment and CBF landscape

MPC-EDCBF

MPC-TCCBF proposed

Snapshots at 4-second intervals · red disc: moving obstacle · green: reference path · play speed ×4 CBF visualization · side view (left) and top view (right)

Experiment ③ · Overtaking results

MPC-TCCBF outperforms MPC-EDCBF

Overtaking experiment: trajectories of MPC-TCCBF (arrives at 40 s) and MPC-EDCBF (arrives at 44 s) with the moving obstacle, plus speed, ED-CBF and TC-CBF values over time

13%↓

Arrival time
46.8 → 40.7 s

53%↓

Speed tracking error
0.133 → 0.063

16%↓

Cross-track error
2.113 → 1.776

Experiment · Summary on the Heron ASV

Faster and more accurate in all scenarios

Scenario	Controller	Arrival time \(t_a\) [s]	Speed error \(e_{\mathrm{speed}}\)	Cross-track error \(e_{\mathrm{cte}}\)
Static	MPC-EDCBF	45.4	0.124	1.765
Static	MPC-TCCBF	41.0	0.040	1.623
Head-on	MPC-EDCBF	42.2	0.097	2.014
Head-on	MPC-TCCBF	38.8	0.051	1.453
Overtaking	MPC-EDCBF	46.8	0.133	2.113
Overtaking	MPC-TCCBF	40.7	0.063	1.776

Short-horizon MPC (\(N=20,\ T_s=0.1\) s) on the Clearpath Heron · desired speed 0.9 m/s

10%↓

Arrival time

56%↓

Speed tracking error

17%↓

Cross-track error

vs MPC-EDCBF · average over the 3 scenarios

Conclusion

TC-CBF for efficient collision avoidance

01

Turning circles in the CBF

The CBF is defined by the clearance between the obstacle and the vehicle's left and right turning circles, so heading and limited turning capability enter the safety constraint.

02

Less conservative, still safe

Only states from which neither turn avoids the obstacle are restricted — MPC-TCCBF avoids unnecessary slowdowns and aggressive turns with a short horizon.

03

Validated on a real ASV

Unicycle simulations and Heron ASV experiments: vs MPC-EDCBF, arrival time 10%↓, speed error 56%↓, cross-track error 17%↓ on average.

FOLLOW-UP WORK Mechatronics 2026COLREGs-compliant avoidance with TC-CBF ICCAS 2026Multimodal trajectory planning with TC-CBF

SAILSafe Autonomy & Intelligence Lab

Thank you

Questions & discussion

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