EV Regenerative Braking and Brake Force Distribution

On a level road, write $W=mg$, $z=a/g$, $k=h/L$ and let $s$ be the static front load fraction. Moment equilibrium gives $N_f=W(s+zk)$ and $N_r=W(1-s-zk)$. Total brake demand is $D=ma=mgz$. Equal adhesion use gives the I-curve: $F_f=zN_f$, $F_r=zN_r$. These balances are derived here; the axle-load and stability treatment is consistent with Limpert, Brake Design and Safety, 3rd edition (2011), and Boerboom (2012), §§1.2.1–1.2.2.

For a lateral demand $y=a_y/g$, available longitudinal adhesion is $\mu_x=\sqrt{\mu^2-y^2}$, assuming lateral force proportional to axle normal load. The front-first envelope is $\max(F_{f,I},D-\mu_xN_r)\le F_f\le\min(D,\mu_xN_f)$. Series energy priority maximizes $\min(F_f,C_f)+\min(D-F_f,C_r)$ over this interval and applies the shared battery limit. Among equally recovering solutions, it chooses the front force closest to the I-curve. Ideal and fixed series modes prescribe the axle split instead.

Each motor envelope is $C_j(v)=f(v)\min(T_jG/r,P_j/v)$ on enabled axles; $f(v)$ is the specified low-speed linear fade. Motor/inverter DC power is $\eta_e v(R_f+R_r)$ and is capped by the BMS terminal acceptance. Stored power is $\eta_e\eta_b v(R_f+R_r)$, where $\eta_b$ is battery charging efficiency; remaining stored-energy headroom limits each interval. Series control replaces axle friction force with regen up to these limits. Parallel control has $H=D-R_f-R_r\ge qD$ and $F_f=\beta H+R_f$. Linear feasibility inequalities provide the exact maximum regenerative contribution for the linked hydraulic split; no force grid is used. An infeasible front-first parallel blend falls back to hydraulic force alone.

For fixed bias $\beta$, incipient lock during a demand ramp occurs at $z_f=\mu_xs/(\beta-\mu_xk)$ when its denominator is positive, and $z_r=\mu_x(1-s)/(1-\beta+\mu_xk)$. These thresholds describe fixed hydraulics, not a speed-varying blended split. The utilization histories assess the selected blend.

The stop has exact kinematics $t_s=v_0/a$, $d_s=v_0^2/(2a)$. An 800-interval midpoint model holds each interval's sampled axle forces and powers. When headroom is exhausted, the interval is split at the saturation time: regen supplies power up to that event and friction supplies the residual thereafter. Both strategies' saturation events are inserted into the energy and force plots, and playback uses the same sampled states and event boundaries. Their sum is checked against $mv_0^2/2$. The controller is an original steady-force implementation informed by Gao and Ehsani's integrated blending framework, not a reproduction of a published controller.

• Level road; constant deceleration and tire friction; both axles remain loaded. All forces are axle totals and braking magnitudes are positive.
• Ideal fixed-ratio gearing maps aggregate generating torque to wheel force. Separate constant motor/inverter and battery charging efficiencies account for regen losses; generating torque and power must be mutually consistent specifications.
• BMS acceptance remains constant through a stop until the entered energy headroom is exhausted. SOC and temperature effects enter through the user-supplied acceptance and headroom, not a universal battery curve.
• Friction brakes can provide any residual axle force instantly. Parallel hydraulics retain the entered fixed front fraction and minimum hydraulic share.
• Lateral demand is allocated proportional to normal load. The friction-circle allowance omits lateral load transfer and tire load sensitivity.
• Only translational kinetic energy is counted. Wheel/motor inertia, aerodynamic and rolling resistance, and grade are excluded.

Use for quasi-static axle allocation and single-stop energy comparisons. Front-first or simultaneous utilization is a lock-order indicator, not a guarantee of yaw stability: actual ABS/ESC action, tire combined-slip behavior, split-friction surfaces, transient pitch and actuator dynamics require a vehicle dynamics model and test validation. Keep ABS and ESC intervention authoritative over regeneration. A fixed rear-biased split is shown for diagnosis and raises a warning; a force demand beyond an axle's grip blocks the results.

The series comparison uses the selected series axle rule; parallel always uses linked hydraulics. Thus parallel can recover more than series constrained to the I-curve. Maximum-regen series has a wider admissible force set than the parallel strategy when both satisfy the same grip and front-first conditions. Infeasible parallel blending falls back to hydraulics and may still have a rear-first tendency; inspect its comparison-table lock tendency and peak grip use.

Energy and force use 800 base intervals with midpoint states, plus explicit battery-saturation events and a zero-force rest point. Playback holds the representative force/power state of its interval and switches regeneration off at the saturation event; energy grows with that same sampled power. The velocity used in power calculations is the interval's midpoint velocity, while wheel position uses the exact stop kinematics. Time resolution is one eight-hundredth of the stopping time; the saturation time is resolved within that sampled-power model. The limiter verdict names a binding constraint; when the full demanded force is regenerated with spare motor and battery capacity it reports that braking demand is satisfied. The schematic shows wheel motion and active regeneration paths; body dimensions are illustrative, while axle spacing and wheel radius follow the inputs. Reference lines are cropped above the static rear axle weight to keep the force diagram readable. Fixed-bias lock boundaries may fall outside the plotted range; a missing front threshold means no finite threshold within this force model.

Regulatory context: UN ECE R13-H, Revision 4 (01 series), with amendments listed by UNECE as consulted on 1 November 2026. No regulatory tables or acceptance bands are reproduced or implemented. Check against the standard in force; these results do not establish regulatory compliance.