se3.scad

use <linalg.scad>
use <so3.scad>

function combine_se3_exp(w, ABt) = construct_Rt(rodrigues_so3_exp(w, ABt[0], ABt[1]), ABt[2]);

// [A,B,t]
function se3_exp_1(t,w) = concat(
	so3_exp_1(w*w),
	[t + 0.5 * cross(w,t)]
);

function se3_exp_2(t,w) = se3_exp_2_0(t,w,w*w);
function se3_exp_2_0(t,w,theta_sq) = 
se3_exp_23(
	so3_exp_2(theta_sq), 
	C = (1.0 - theta_sq/20) / 6,
	t=t,w=w);

function se3_exp_3(t,w) = se3_exp_3_0(t,w,sqrt(w*w)*180/PI,1/sqrt(w*w));

function se3_exp_3_0(t,w,theta_deg,inv_theta) = 
se3_exp_23(
	so3_exp_3_0(theta_deg = theta_deg, inv_theta = inv_theta),
	C = (1 - sin(theta_deg) * inv_theta) * (inv_theta * inv_theta),
	t=t,w=w);

function se3_exp_23(AB,C,t,w) = 
[AB[0], AB[1], t + AB[1] * cross(w,t) + C * cross(w,cross(w,t)) ];

function se3_exp(mu) = se3_exp_0(t=take3(mu),w=tail3(mu)/180*PI);

function se3_exp_0(t,w) =
combine_se3_exp(w,
// Evaluate by Taylor expansion when near 0
	w*w < 1e-8 
	? se3_exp_1(t,w)
	: w*w < 1e-6
	  ? se3_exp_2(t,w)
	  : se3_exp_3(t,w)
);

function se3_ln(m) = se3_ln_to_deg(se3_ln_rad(m));
function se3_ln_to_deg(v) = concat(take3(v),tail3(v)*180/PI);

function se3_ln_rad(m) = se3_ln_0(m, 
	rot = so3_ln_rad(rotation_part(m)));
function se3_ln_0(m,rot) = se3_ln_1(m,rot,
	theta = sqrt(rot*rot));
function se3_ln_1(m,rot,theta) = se3_ln_2(m,rot,theta,
	shtot = theta > 0.00001 ? sin(theta/2*180/PI)/theta : 0.5,
	halfrotator = so3_exp_rad(rot * -.5));
function se3_ln_2(m,rot,theta,shtot,halfrotator) =
concat( (halfrotator * translation_part(m) - 
	(theta > 0.001 
	? rot * ((translation_part(m) * rot) * (1-2*shtot) / (rot*rot))
	: rot * ((translation_part(m) * rot)/24)
	)) / (2 * shtot), rot);

__se3_test = [20,-40,60,-80,100,-120];
echo(UNITTEST_se3=norm(__se3_test-se3_ln(se3_exp(__se3_test))) < 1e-8);