Robust　nonlinear　regression　frequently　arises　in　data　analysis　that　is　affected　by　outliers　in　various　application　fields　such　as　system　identification,　signal　processing,　and　machine　learning.　However,　it　is　still　quite　challenge　to　design　an　efficient　algorithm　for　such　problems　due　to　the　nonlinearity　and　nonsmoothness.　Previous　researches　usually　ignore　the　underlying　structure　presenting　in　the　such　nonlinear　regression　models,　where　the　variables　can　be　partitioned　into　a　linear　part　and　a　nonlinear　part.　Inspired　by　the　high　efficiency　of　variable　projection　algorithm　for　solving　separable　nonlinear　least　squares　problems,　in　this　paper,　we　develop　a　robust　variable　projection　(RoVP)　method　for　the　parameter　estimation　of　separable　nonlinear　regression　problem　with　$L_{1}$　norm　loss.　The　proposed　algorithm　eliminates　the　linear　parameters　by　solving　a　linear　programming　subproblem,　resulting　in　a　reduced　problem　that　only　involves　nonlinear　parameters.　More　importantly,　we　derive　the　Jacobian　matrix　of　the　reduced　objective　function,　which　tackles　the　coupling　between　the　linear　parameters　and　nonlinear　parameters.　Furthermore,　we　observed　an　intriguing　phenomenon　in　the　landscape　of　the　original　separable　nonlinear　objective　function,　where　some　narrow　valleys　frequently　exist.　The　RoVP　strategy　can　effectively　diminish　the　likelihood　of　the　algorithm　getting　stuck　in　these　valleys　and　accelerate　its　convergence.　Numerical　experiments　confirm　the　effectiveness　and　robustness　of　the　proposed　algorithm.　IEEE