Table 2: The performance of single-agent algorithms.  $\bar{J}^{R}$  stands for normalized reward from PPO's performance, and  $\bar{J}^{C}$  signifies normalized cost relative to cost_limit. The  $\uparrow$  indicates higher rewards are better, while the  $\downarrow$  indicates lower costs (when beyond the threshold of 1.00) are better. Gray and Black depicts breach and compliance with the cost_limit, while Green represents the optimal policy, maximizing reward within safety constraints.
PPO	PPO-Lag	TRPO-Lag	CPPO-PID	RCPO	CPO	PCPO	CUP	FOCOPS
Safety Navigation	JR↑	JC↓	JR↑	JC↓	JR↑	JC↓	JR↑	JC↓	JR↑	JC↓	JR↑	JC↓	JR↑	JC↓	JR↑	JC↓	JR↑	JC↓
ANTBUTTON1	1.00	4.42	0.09	0.86	0.23	1.95	0.10	0.70	0.16	2.07	0.12	4.01	0.03	1.01	0.03	0.17	0.01	0.46
ANTCIRCLE1	1.00	16.81	0.79	2.56	0.65	1.05	0.69	1.90	0.63	1.04	0.47	1.07	0.28	1.87	0.60	0.82	0.02	1.22
ANTGOAL1	1.00	1.81	0.26	0.94	0.25	0.74	0.47	1.94	0.29	0.78	0.19	0.55	0.09	0.42	0.34	1.33	0.09	0.67
ANTPUSH1	1.00	1.90	0.13	0.00	0.30	0.00	0.13	0.00	0.33	0.00	0.17	0.00	0.07	0.00	0.20	0.00	-0.30	0.03
CARBUTTON1	1.00	16.09	0.01	0.47	-0.04	1.08	-0.10	0.40	-0.19	1.73	0.08	1.75	0.02	1.90	0.04	5.50	0.04	1.21
CARCIRCLE1	1.00	8.42	0.81	0.82	1.69	2.77	1.61	1.79	1.70	3.11	1.67	3.13	1.41	1.99	0.76	1.04	0.84	1.12
CARGOAL1	1.00	2.38	0.43	0.39	0.82	1.09	0.03	2.47	0.55	0.86	0.78	1.63	0.61	1.42	0.19	0.63	0.52	0.93
CARPUSH1	1.00	7.16	0.46	0.78	1.38	0.70	0.03	0.47	1.11	1.42	0.83	1.14	0.64	2.36	0.32	0.95	0.29	0.36
DOGGOBUTTON1	1.00	7.57	0.01	0.03	0.00	1.27	0.01	0.07	0.01	0.09	0.00	0.15	0.00	0.25	0.02	0.45	0.06	3.68
DOGGOCIRCLE1	1.00	33.14	0.77	0.46	0.67	1.37	0.82	2.16	0.55	1.32	0.66	1.22	0.31	0.55	0.80	2.04	0.73	4.49
DOGGOGAOL1	1.00	2.28	0.05	0.00	0.18	0.69	0.00	0.00	0.16	2.08	0.30	0.50	0.00	0.00	0.00	0.90	0.04	1.27
DOGGPUSH1	1.00	1.31	0.09	0.00	0.53	0.78	0.32	0.44	0.54	1.55	0.46	0.00	0.36	0.00	0.30	0.68	0.64	3.40
POINTBUTTON1	1.00	6.06	0.22	1.32	0.27	1.29	0.00	0.84	0.12	1.13	0.12	1.61	0.08	2.19	0.18	1.26	0.25	1.53
POINTCIRCLE1	1.00	8.10	0.86	0.93	1.67	1.35	1.72	2.09	1.66	1.42	1.69	1.74	1.33	2.26	0.82	0.62	0.84	0.89
POINTGAOL1	1.00	1.93	0.47	1.50	0.72	0.91	0.31	1.05	0.53	0.99	0.78	1.10	0.71	0.82	0.46	0.73	0.56	1.32
POINTPUSH1	1.00	2.31	0.98	1.33	0.85	1.00	0.35	0.35	5.30	0.94	2.22	0.80	1.72	1.25	2.32	0.80	1.13	2.51
RACECARBUTTON1	1.00	13.73	-0.01	1.94	-0.02	1.77	-0.16	2.06	-0.07	1.19	0.00	2.44	0.02	1.82	0.00	5.23	-0.10	3.37
RACECARCIRCLE1	1.00	15.87	0.83	1.90	0.80	2.18	0.58	1.33	0.83	2.07	0.79	0.81	0.22	2.87	0.74	3.53	0.77	2.11
RACECARGOAL1	1.00	4.26	0.26	0.51	1.19	0.77	-0.04	1.07	0.88	0.83	1.18	2.58	0.33	0.24	0.13	1.22	0.31	0.62
RACECARPUSH1	1.00	2.34	-0.40	0.00	0.74	1.79	-0.84	2.87	0.58	1.92	0.94	0.13	-0.16	0.18	-0.06	3.79	0.30	2.04
Safety Velocity	JR↑	JC↓	JR↑	JC↓	JR↑	JC↓	JR↑	JC↓	JR↑	JC↓	JR↑	JC↓	JR↑	JC↓	JR↑	JC↓	JR↑	JC↓
ANTVEL	1.00	38.33	0.54	0.22	0.53	0.15	0.51	0.41	0.52	0.56	0.52	0.56	0.38	0.41	0.55	0.94	0.55	0.60
HALFCHEETAHVEL	1.00	36.77	0.44	0.00	0.43	1.01	0.48	0.04	0.36	0.56	0.40	0.23	0.25	0.63	0.40	0.17	0.42	0.12
HOPPERVEL	1.00	22.00	0.58	0.89	0.59	0.71	0.73	0.44	0.58	0.59	0.73	0.48	0.65	0.51	0.73	0.21	0.66	0.30
HUMANOIDVEL	1.00	38.42	0.72	0.76	0.72	2.38	0.73	0.00	0.68	0.82	0.71	0.01	0.64	0.01	0.68	0.80	0.71	0.93
SWIMMERVEL	1.00	6.61	0.57	1.11	0.66	0.84	0.91	0.92	0.54	0.90	0.51	0.82	0.50	0.69	0.59	0.96	0.47	1.30
WALKER2DVEL	1.00	36.11	0.44	0.20	0.51	0.77	0.27	0.36	0.49	0.15	0.39	0.81	0.27	0.71	0.44	0.18	0.50	0.16